How to Integrate Using U-Substitution (NancyPi) @NancyPi

NancyPi | How to Integrate Using U-Substitution (NancyPi) @NancyPi | Uploaded August 2018 | Updated October 2024, 3 hours ago.
MIT grad shows how to do integration using u-substitution (Calculus). To skip ahead: 1) for a BASIC example where your du gives you exactly the expression you need in order to substitute, skip to time 1:30. 2) For an example where you have to REARRANGE THE DU by multiplying or dividing because the du has a different number or sign than what you need, skip to time 8:21. 3) For one where you have to REARRANGE THE U by subtracting or adding because the du expression cannot give you the expression you need, skip to 17:15. 4) For u-substitution with TRIG (SIN/COS) and the power rule, skip to 22:35. Nancy formerly of MathBFF explains the steps.

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For my video on the BASICS of INTEGRATION, jump to: youtu.be/e1nxhJQyLYI

With all u-substitution integration problems:

The FIRST STEP is to pick your "u". The best choice is usually the longer x-expression that is inside a power or a square root or the denominator, etc (in an "inside function"). Set u equal to this x-expression.

The SECOND STEP is to find "du" by taking the derivative of the u expression with respect to x. For instance, if you have u=3x+2, your du would then be du=3dx. **NOTE: Remember to include "dx" at the end of your du differential expression.

The THIRD STEP is to substitute u and du into the integral everywhere in place of x and dx. **NOTE: if your du does not perfectly match what you need in order to completely substitute before integrating, you must rearrange the du, or sometimes rearrange the u, in order to fully substitute before integrating. For an example of each, see example #2 (time: 8:21) and example #3 (time: 17:15).

The FOURTH STEP is to integrate, remembering to add "+ C" at the end since you integrated an indefinite integral (no limits). The du goes away when you integrate.

The LAST STEP is to "back-substitute" by replacing everywhere u appears with the x-expression that you chose u to be.

For more of my math videos, check out: nancypi.com

$Synthetic Division... How? (NancyPi) MIT grad shows how to do synthetic division, a shortcut for long division. Its a fast way of dividing polynomials, if youre dividing one polynomial by a linear expression like x+1 or x-3. To skip ahead: 1) For how to know WHICH NUMBERS to put in the corner of the division box and the first row, skip to time 0:22. 2) For the synthetic division STEPS of carrying down coefficients, multiplying, and adding, skip to 1:12. 3) For how to WRITE the FINAL ANSWER and how to WRITE the REMAINDER (if the last number you get is not zero, there is a remainder), skip to 3:37. 4) For what to do if you have a MISSING TERM and putting a zero as its placeholder, skip to 5:04. Nancy formerly of MathBFF explains the steps. For my video on how to do LONG DIVISION of polynomials instead, jump to: https://youtu.be/RPXMBIFG_W4 For my video on FACTORING to solve for the zeros of a polynomial, jump to: https://youtu.be/Z5MnP9da4EM Follow Nancy on Instagram: https://instagram.com/nancypi Twitter: https://twitter.com/nancypi SYNTHETIC DIVISION is often faster than long division for dividing polynomials, if the polynomial youre dividing by is x plus a number or x minus a number. It can be used to factor a polynomial and to find the zeros or roots of a polynomial. HOW TO DO synthetic division of polynomials: First draw a little corner symbol. Inside the corner, write the number that makes your denominator equal to zero. If your denominator is x+1, you would write -1 in the corner. Then write the coefficients of your top polynomial in a row, to the right of the corner symbol. Leave space for another row and draw a horizontal line beneath that. Here are the steps to repeat for synthetic division, to carry down coefficient numbers, multiply, and add: 1) Drop down the first coefficient number. 2) Multiply this number by the corner constant. 3) Write the product in the second column, second row spot. 4) Add the two numbers in the second column, and write the sum below the line in the second column. 5) Repeat those steps until you have written a number in the final column spot below the line. 6) Write your new polynomial answer using this bottom row of numbers. The first number is the leading coefficient of your polynomial and is the coefficient of an x-term that is one degree less than your original polynomial. The second number is the coefficient of the second term, etc. For example, if the numbers are 1 4 3 2, the new polynomial is 1x^2 + 4x + 3 + 2/(x+1). Notice the last number is part of the remainder, so you write that number, 2, over the original divisor (x+1) as a fraction, for the remainder term. Sometimes there is no remainder in synthetic division examples. If the last number you get is zero, in your final division numbers, then there is no remainder. Your final answer will just have a polynomial and no fraction term added at the end. Synthetic division is a way of factoring polynomials: if there is no remainder when you finish, you have found a true factor of the original polynomial. In the example above, if the remainder had been zero instead, the original polynomial would factor into (x+1)(x^2 + 4x + 3). Note: also watch out for a missing term in your original polynomial: if there is no x-squared term, or no x term, etc, then in order to get the right answer for synthetic division, you will need to make sure to put a zero number in your first row of coefficient numbers, as a placeholder for that missing term. For more rational function and rational expression math examples in my algebra, algebra 2, precalculus, and college algebra mathematics videos, and for more practice problems, check out: http://nancypi.com$ $How to Simplify Radicals (NancyPi) MIT grad shows how to simplify radical expressions, specifically square root expressions, into their simplest form (Simplified Radical Form or SRF Form). To skip ahead: 1) for a PERFECT SQUARE under the root like sqrt(16) skip to time 1:29. 2) for a SMALL number under the root that is NOT a perfect square like sqrt(32), skip to time 2:45. 3) for a LARGE number under the root, such as sqrt(343), skip to 5:14. 4) for a PRODUCT, meaning a number times a square root like 5*sqrt(54) skip to 7:30. 5) for a QUOTIENT, meaning a number divided by a square root like 4/sqrt(2), and where you have to RATIONALIZE the denominator, skip to 10:10. and 6) for a more complex QUOTIENT example like (4+sqrt(3))/(5-sqrt(3)) where you have to multiply by the CONJUGATE to simplify, skip to 12:36. Nancy formerly of MathBFF explains the steps. Follow me on Instagram! https://instagram.com/mathnancy Follow me on Twitter: https://twitter.com/nancypi For all simplifying radical problems: 1) First try to find a perfect square number that divides evenly into the number under the square root. Perfect square numbers include 4, 9, 16, 25, 36, 49, etc. because they are squares of the numbers 2, 3, 4, 5, 6, 7, etc., respectively. 2) Separate the root expression into the two numbers you found as factors, with each number now under its own square root symbol. For example, sqrt(12) = sqrt(4) times sqrt(3). 3) Keep simplifying. The square root of a perfect square becomes just a number, without the radical square root symbol. sqrt(4)*sqrt(3) becomes 2*sqrt(3). 4) Check to see if anymore simplifying can be done, either by combining constants or finding another perfect square number that divides evenly into your factors. If no more simplifying can be done, you have your final simplified radical form (SRF) answer, 2*sqrt(3). If there is a square root anywhere in the DENOMINATOR, you must rationalize the denominator, in other words, somehow get rid of the root in the bottom, in order for the expression to be considered simplest. For example, if you have 4/sqrt(2) you must use the trick of simplifying by multiplying the expression by sqrt(2)/sqrt(2). This will clear the square root in the bottom. If you multiply straight across, you would get 4*sqrt(2)/2, which simplifies again to 2*sqrt(2). If you have a more complicated fraction with a square root in the denominator, like (4+sqrt(3))/(5-sqrt(3)), you will need to use the conjugate to simplify. This means multiplying top and bottom by the expression in the denominator, but with the middle sign flipped. Here you would multiply by (5+sqrt(3))/(5+sqrt(3)), multiply straight across on top and bottom using FOIL-ing and distribution. Some terms will cancel in the bottom, leaving you with a simpler denominator that has no square roots. For more of my math videos, check out: http://nancypi.com$ $How to Find the Domain of Any Function (NancyPi) MIT grad shows a surefire way to find the domain of any function. To skip ahead: 1) For POLYNOMIAL only, skip to time 0:45. 2) For FRACTION only, skip to time 1:44. 3) For SQUARE ROOT only, skip to 5:08. 4) For SQUARE ROOT ON BOTTOM of Fraction, skip to 8:44. 5) For SQUARE ROOT ON TOP of Fraction, skip to 10:09. Nancy formerly of MathBFF explains the steps. For how to SOLVE QUADRATIC equations by FACTORING, jump to: https://youtu.be/Z5MnP9da4EM For how to SOLVE INEQUALITIES, jump to: https://youtu.be/ONuIIvgkE3c Follow Nancy on Instagram! https://instagram.com/nancypi Twitter: https://twitter.com/nancypi For Domain, you just need to watch out for Square Roots and/or Fractions! If your function is... 1) a POLYNOMIAL (no square roots or fractions), i.e. f(x) = x^2 + 3x + 1, domain is all real numbers. 2) a FRACTION (w/ no square root), i.e. f(x) = (2x+1)/(x^2+5x+6). Set bottom not equal to zero. Solve for x. 3) a SQUARE ROOT (w/ no fraction), i.e. f(x) = sqrt(x+1). Set part under root greater than or equal to zero. 4) a FRACTION w/ A SQUARE ROOT ON BOTTOM, i.e. f(x) = (x^2+2x+3)/(sqrt(x+1)). Set part under root greater than zero. Solve the inequality. NOTE THE DIFFERENCE: You must set it greater than zero, not greater than OR EQUAL TO zero because we cant have zero (or square root of zero) in the denominator. 5) a FRACTION w/ A SQUARE ROOT ON TOP, i.e. f(x) = (sqrt(x+1))/(x^2+4), i) Set the bottom not equal to zero & solve; ii) Set whats under the root greater than or equal to zero & solve; iii) Take intersection of your two solutions. For more of my math videos, check out: http://nancypi.com$

$How to Solve Inequalities (NancyPi) MIT grad explains solving inequalities. This video focuses on solving linear inequalities. It shows when to switch the sign of the inequality, if you divide or multiply by a negative number, and is an introduction to how to solve inequalities in algebra. To skip ahead: 1) For a basic example of SOLVING AN INEQUALITY, skip to time 00:34. 2) For when you have to FLIP THE INEQUALITY SIGN when DIVIDING BY A NEGATIVE number, skip to 1:26. 3) For when you need to reverse the inequality symbol because youre MULTIPLYING BY A NEGATIVE number, skip to 2:42. The problems here are multi-step inequalities, or two-step inequalities. Nancy formerly of MathBFF explains the steps. Follow Nancy on Instagram: https://instagram.com/nancypi Twitter: https://twitter.com/nancypi Algebraic equations and inequalities: what is inequality math? What does inequality mean? An inequality is a number sentence that uses an inequality symbol instead of an equality symbol (=), which means it has a greater than sign, less than sign, greater than or equal to sign, or less than or equal to sign, instead of the normal equal sign in an equation. One linear inequality definition is: an inequality that involves a linear algebraic expression and inequality signs. How to find the solution of an inequality algebra problem: 1) A simple, BASIC LINEAR INEQUALITY example you might see on a solving inequalities worksheet or problem set is the expression 2x + 3 [less than or equal to] 11. You can solve for x just like you would for an equality, or normal equation, by trying to get x alone. This is an example of two-step inequalities, or multi-step inequalities. The first step is to subtract the constant 3 from both sides, in order to move it to the right hand side, so that you have 2x [less than or equal to] 8. The second step is to get x alone on the left side by dividing both sides by 2 so that your answer for the inequality is x [less than or equal to] 4. In this case, solving the inequality used the exact same steps you would use to solve an equal sign equation. Sometimes the steps are different from solving normal math equations. Solving for x is the same as solving a linear equality (-2x + 3 = 11) except that if you ever have to divide by a negative number or multiply by a negative number while solving for x, you must flip the sign, or switch the direction of your inequality symbol less than becomes greater than, for example). Here are the two cases of inequality problems WHEN YOU NEED TO REVERSE THE SIGN of the inequality because youre multiplying or dividing both sides of the inequality by a negative number: 2) DIVIDING EXAMPLE: For an inequality like -2x + 3 [less than or equal to] 11, after moving the constant 3 to the right side, you will need to divide both sides by negative 2 in order to get x alone on the left side for the solution. WHEN YOU DIVIDE BY A NEGATIVE NUMBER, you need to reverse the direction of the inequality symbol in your expression. In this example, after you divide out -2, you get x [greater than or equal to] -4 (instead of x less than or equal to -4) for the final solution. This example was also a two-step inequality problem. 3) MULTIPLYING EXAMPLE: For examples like (-1/2)x + 4 [greater than] 7, when youve moved the constant to the right side and have the inequality (-1/2)x [greater than] 3, the fastest way to get x alone is to clear the fraction by multiplying both sides by -2. This will cancel the 2 in the denominator and remove the negative sign in front of x on the left side. But WHEN YOU MULTIPLY BY A NEGATIVE, you have to flip the direction of the inequality symbol so that your answer is x [less than] -6 instead of x [greater than] -6. This problem was also a multi-step inequality. Note on inequality rules: you dont have to switch the symbol if youre adding a negative number or subtracting a negative number: switching the inequality sign is only for when youre multiplying or dividing by a negative number. For quadratic inequalities examples, compound inequalities, inequality math, inequality algebra, inequality equations, inequality symbols, inequality notation, and more algebra solver, equation solver, and inequality calculator practice problems that come up on worksheets, as well as more algebra, algebra 2, and college algebra mathematics videos, check out: http://nancypi.com$

$Derivatives... How? (NancyPi) MIT grad shows how to find derivatives using the rules (Power Rule, Product Rule, Quotient Rule, etc.). To skip ahead: 1) For how and when to use the POWER RULE, constant multiple rule, constant rule, and sum and difference rule, skip to time 0:22. 2) For the PRODUCT RULE, skip to 7:36. 3) For the QUOTIENT RULE, skip to 10:53. For my video on the CHAIN RULE for finding derivatives: https://youtu.be/H-ybCx8gt-8 For my video on the DEFINITION of the derivative: https://youtu.be/-ktrtzYVk_I Nancy formerly of MathBFF explains the steps. For more of the QUOTIENT RULE and a shortcut to remember the formula, jump to my video at: https://youtu.be/jwuiVb84Xx4 For Follow Nancy on Instagram: https://instagram.com/nancypi Twitter: https://twitter.com/nancypi What is the derivative? Its a function that gives you the instantaneous rate of change at each point of another function. You can calculate the derivative with the definition of the derivative (using the limit), but the fastest way to find the derivative is with shortcuts such as the Power Rule, Product Rule, and Quotient Rule: 1) POWER RULE: If the given equation is a polynomial, or just a power of x, then you can use the Power Rule. For a term thats just a power of x, such as x^4, you can get the derivative by bringing down the power to the front of the term as a coefficient and decreasing the x power by 1. For example, for x^4, the derivative is 4x^3. If you have many terms added or subtracted together, and if they are powers of x, you can use the Power Rule on each term (by the Sum and Difference Rules). NOTE: The derivative of a constant, just a number, is always 0 (that is the Constant Rule). Also, if you have a term that is a constant multiplied in the front of the term, like 2x^3, you can keep the constant and differentiate the rest of the term. In this example, you keep the 2 and take the derivative of x^3, which is 3x^2, so the derivative of the term 2x^3 is 2*3x^2, or 6x^2. ANOTHER NOTE:You can use the same power rule method for fractional or negative powers, but be careful... for negative powers, it works as long as x is not 0, and for fractional/rational powers, if the power is less than 1, your derivative wont be defined at x = 0. 2) PRODUCT RULE: If your equation is not a polynomial but instead has the overall form of one expression multiplied by another expression, then you can use the Product Rule. The Product Rule says that the derivative of two functions multiplied together is equal to the first function times the derivative of the second function, plus the second function times the derivative of the first function. 3) QUOTIENT RULE: If your equation has the overall form of one expression divided by another expression, then you can use the Quotient Rule. The Quotient Rule says that the derivative of one function divided by another (a quotient) is equal to the bottom function times the derivative of the top bottom minus the top function times the derivative of the bottom function, all divided by the bottom function squared. This is true as long as the bottom function is not equal to 0. For more of my math videos, check out: http://nancypi.com$

How to do Long Division with Polynomials (NancyPi)

$Power Rule... How? (NancyPi) MIT grad shows how to find the derivative using the Power Rule, one of the derivative rules in calculus. It is a shortcut for taking derivatives of polynomial functions with powers of x. To skip ahead: 1) For HOW and WHEN to use the power rule, skip to time 0:22. 2) For how to use the power rule when you have a FRACTIONAL or NEGATIVE POWER, skip to 5:22. For my video on the other differentiation rules, PRODUCT RULE and QUOTIENT RULE, skip to https://youtu.be/QqF3i1pnyzU?t=456 Nancy formerly of mathbff explains the steps: For how to differentiate using the formal LIMIT DEFINITION of the derivative instead, jump to https://youtu.be/-ktrtzYVk_I?t=628 Support Nancy on Patreon: https://www.patreon.com/nancypi Follow Nancy on Instagram: https://instagram.com/nancypi Follow Nancy on Twitter: https://twitter.com/nancypi HOW and WHEN to use the POWER RULE: If the given equation is a polynomial, or just a power of x, then you can use the Power Rule. For a term thats just a power of x, such as x^4, you can get the derivative by bringing down the power to the front of the term as a coefficient and decreasing the x power by 1. For example, for x^4, the derivative is 4x^3. If you have many terms added or subtracted together, and if they are powers of x, you can use the Power Rule on each term (by the Sum and Difference Rules). NOTE: The derivative of a constant, just a number, is always 0 (that is the Constant Rule). Also, if you have a term that is a constant multiplied in the front of the term, like 2x^3, you can keep the constant and differentiate the rest of the term (Constant Multiple Rule). In this example, you keep the 2 and take the derivative of x^3, which is 3x^2, so the derivative of the term 2x^3 is 2*3x^2, or 6x^2. ANOTHER NOTE: You can use the same power rule method for fractional or negative powers, but be careful... for negative powers, it works as long as x is not 0, and for fractional/rational powers, if the power is less than 1, your derivative wont be defined at x = 0. The derivative is a function that gives you the instantaneous rate of change at each point of another function. You can calculate the derivative with the definition of the derivative (using the limit, see https://youtu.be/-ktrtzYVk_I?t=628), but the fastest way to find the derivative is with shortcuts such as the Power Rule, Product Rule, and Quotient Rule. For my video on the CHAIN RULE for finding derivatives, jump to https://youtu.be/H-ybCx8gt-8 For more math help and videos, check out: http://nancypi.com Editor: Miriam Nielsen of zentouro https://www.youtube.com/zentouro Director: Kristopher Knight https://kristopherknight.com$