Integrals > U Substitution

**Contents:**

- Overview and Basic Example
- U Substitution for Trigonometric Functions
- U Substitution for Definite Integrals
- 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(x^{2} + 3)^{70}. Recognizing that if you differentiate x^{2} + 3, you get 2x, is the key to successful u substitution.

## Example Problem

**Example problem 1:** Integrate 2x(x^{2} + 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 (x^{2}) with u makes the function look more familiar for integrating:

- u = x
^{2}+ 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”**:

## 2. U Substitution for Trigonometric Functions

U substitution is one way you can find integrals for trigonometric functions.

## 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 =
^{5}⁄_{4}∫ 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:

^{5}⁄_{4}∫ sec u tan u du =^{5}⁄_{4}sec u + C

Step 5: **Re-substitute for u**:

^{5}⁄_{4}sec u + C =^{5}⁄_{4}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!*

## 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:

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:

Step 4: **Integrate the function **from Step 3:

Step 5: **Evaluate at the given limits**:

*That’s it!*

## 4. U Substitution for Exponential Functions

**Example question**: Find the integral for the exponential function ^{ex + 1}⁄_{ex} using u substitution.

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

- ∫
^{ex + 1}⁄_{ex}= - ∫(
^{ex}⁄_{ex}+^{1}⁄_{ex}) = - ∫(1 + e
^{ – x})dx

Step 2: **Split the function** into separate parts:

- ∫(1 + e
^{ – x})dx = - ∫1dx + ∫e
^{ – x}dx

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 + ∫e
^{u}(-1)du

Step 6: **Solve the integrals:**

- ∫1dx + ∫e
^{u}(-1)du = x – e^{u}+ C

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

- x – e
^{u}+ C = x – e^{-x}+ C

*That’s it!*

**CITE THIS AS:**

**Stephanie Glen**. "U Substitution (Reverse Chain Rule)" From

**CalculusHowTo.com**: Calculus for the rest of us! https://www.calculushowto.com/u-substitution/

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