Calculus How To

U Substitution (Reverse Chain Rule)

Integrals > U Substitution

Contents:

  1. Overview and Basic Example
  2. U Substitution for Trigonometric Functions
  3. U Substitution for Definite Integrals
  4. U Substitution for Exponential Functions

1. Overview and Basic Example

U substitution (also called integration by substitution or u substitution) takes a rather complicated integral and turns it—using algebra and an auxiliary function or two—into integrals you can recognize and easily integrate.

U substitution requires strong algebra skills and knowledge of rules of differentiation. Why? Because you’ll need to be able to look at the integral and see where a little algebra might get the form into one you can easily integrate—and as integration is really reverse-differentiation, knowing your rules of differentiation will make the task much easier.

For example, the following example problem uses the integral 2x(x2 + 3)70. Recognizing that if you differentiate x2 + 3, you get 2x, is the key to successful u substitution.

Example Problem

Example problem 1: Integrate 2x(x2 + 3)70 using integration by substitution.

Step 1: Choose a term to substitute for u. Pick a term that when you substitute u in, it makes it easily to integrate. In this example, replacing (x2) with u makes the function look more familiar for integrating:

  • u = x2 + 3

Step 2: Rewrite the function with the new function “u” from Step 1.


I pulled the constant out in front here. If you don’t understand why, you can find intermediate steps with algebraic formulas on Symbolab’s integral calculator.

Step 3: Apply the power function rule for integration:


Which gives you:


Step 4: Substitute “u” back in and simplify:


Step 5: Add a “C”:


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2. U Substitution for Trigonometric Functions

U Substitution Trigonometric Functions
U substitution is one way you can find integrals for trigonometric functions. W

U Substitution Trigonometric Functions: Examples

Example problem #1: Integrate ∫sin 3x dx.

Step 1: Select a term for “u.” Look for substitution that will result in a more familiar equation to integrate. Substituting u for 3x will leave an easier term to integrate (sin u), so:

  • u = 3x

Step 2: Differentiate u:

  • du = 3 dx

Or (rewriting using algebra—necessary because you need to replace “dx”, not 3 dx:
⅓ du = dx

Step 3: Replace all forms of x in the original equation:

  • Substituting for u: ∫ sin 3x dx = ∫ sin u dx
  • Substituting for dx: ∫ sin u dx = ∫ sin u ⅓ du

Step 4: Rewrite, bringing the constant in front of the integral symbol (so that you can easily integrate):

  • ∫ sin u ⅓ du = ⅓ ∫ sin u du

Step 5: Integrate using the usual rules of integration:

  • ⅓ ∫ sin u du = ⅓ (-cos u) + C = -⅓ cos u + C

Step 6: Re-substitute for u:
“u” is left in the equation, so:

  • ⅓ cos u + C = ⅓ cos 3x + C

That’s it!

Example problem #2: Integrate ∫ 5 sec 4x dx

Step 1: Pick a term to substitute for u:

  • u = 4x

Step 2: Differentiate, using the usual rules of differentiation.

  • du = 4 dx
  • ¼ du = dx (using algebra to rewrite, as you need to substitute dx on its own, not 4x)

Step 3: Substitute u and du into the equation:

  • ∫ 5 sec 4x tan 4x dx = 5 ∫ sec u tan u ¼ du =
  • 54 ∫ sec u tan u du

Step 4: Integrate, using the usual rules of integration. For this problem, integrate using the rule D(sec x) = sec x tan x:

  • 54∫ sec u tan u du = 54 sec u + C

Step 5: Re-substitute for u:

  • 54 sec u + C = 54 sec 4x + C

Tip: If you don’t know the rules by heart, compare your function to the general rules of integration and look for familiar looking integrands before you attempt to substitute anything for u.

That’s all there is to U Substitution for Trigonometric Functions!

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U Substitution for Definite Integrals

In general, a definite integral is a good candidate for u substitution if the equation contains both a function and that function’s derivative. When evaluating definite integrals, figure out the indefinite integral first and then evaluate for the given limits of integration.

Example problem: Evaluate:
u-substitute-for-definite-integrals

Step 1: Pick a term for u. Choose sin x for this example problem, because the derivative is cos x.
u = sin x.

Step 2: Find the derivative of u:

  • du = cos x dx

Step 3: Substitute u and du into the function:
u-substitute-for-definite-integrals-2

Step 4: Integrate the function from Step 3:
u-substitute-for-definite-integrals-3

Step 5: Evaluate at the given limits:
u-substitute-for-definite-integrals-4

That’s it!

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4. U Substitution for Exponential Functions

Example question: Find the integral for the exponential function ex + 1ex using u substitution.

Step 1: Rewrite your function using algebra to get it in a form where you can easily find an integral:

  • ex + 1ex =
  • ∫(exex + 1ex) =
  • ∫(1 + e – x)dx

Step 2: Split the function into separate parts:

  • ∫(1 + e – x)dx =
  • ∫1dx + ∫e – xdx

Step 3: Pick u and find the derivative of u. For this example, pick “-x” in e-x:

  • u = -x
  • du = -1·dx

Step 4: Find a way to remove the symbol dx using your second substitution in Step 3. Using algebra:

  • du =-1·dx, so
  • -1du = dx

Step 5: Substitute the “u”, and “du” from Steps 3 and 4 into the equation.

  • ∫1dx + ∫eu(-1)du

Step 6: Solve the integrals:

  • ∫1dx + ∫eu(-1)du = x – eu + C

Step 7: Re-substitute your terms back into the function. u = -x, so:

  • x – eu + C = x – e-x + C

That’s it!

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