Integration by U-Substitution for AP Calculus

Practice problems for these concepts can be found at: Integration Practice Problems for AP Calculus

In this study guide:

• The U -Substitution Method
• U -Substitution and Algebraic Functions
• U -Substitution and Trigonometric Functions
• U -Substitution and Inverse Trigonometric Functions
• U -Substitutionand Logarithmic and Exponential Functions

The U-Substitution Method

The Chain Rule for Differentiation

F (g(x)) = f(g(x))g '(x), where F ' = f

The Integral of a Composite Function

If f(g(x)) and f ' are continuous and F ' = f, then

f(g(x))g '(x)dx = F(g(x)) + C.

Making a U-Substitution

Let u =g(x), then du = g '(x)dx

f(g(x))g "(x)dx = f (u)du = F(u) + C = F(g(x)) + C.

Procedure for Making a U-Substitution

Steps:

1. Given f(g(x)); let u = g(x).
2. Differentiate: du = g '(x)dx
3. Rewrite the integral in terms of u.
4. Evaluate the integral.
5. Replace u by g(x).
6. Check your result by taking the derivative of the answer.

U-Substitution and Algebraic Functions

Another Form of the Integral of a Composite Function

If f is a differentiable function, then

Making a U-Substitution

Example 1

Evaluate x (x + 1)¹⁰dx.

Step 1.   Let u = x + 1; then x = u – 1.

Step 2.   Differentiate: du = dx.

Step 3.   Rewrite: (u – 1) u¹⁰du = (u¹¹ – u¹⁰)du.

Step 4.   Integrate: .

Step 5.   Replace .

Step 6.   Differentiate and Check:

Example 2

Evaluate .

Step 1.   Let u = x – 2; then x = u + 2.

Step 2.   Differentiate: du = dx.

Step 3.   Rewrite: .

Step 4.   Integrate: .

Step 5.   Replace: .

Step 6.   Differentiate and Check: .

Example 3

Evaluate (2x – 5)^2/3dx.

Step 1.   Let u = 2x – 5.

Step 2.   Differentiate: du = 2dx = dx.

Step 3.   Rewrite:

Step 4.   Integrate:

Step 5.   Replace

Step 6.   Differentiate and Check:

Example 4

Evaluate

Step 1.   Let u = x³ – 8.

Step 2.   Differentiate: du =3x²dx = x²dx.

Step 3.   Rewrite:

Step 4.   Integrate:

Step 5.   Replace

Step 6.   Differentiate and Check:

U-Substitution and Trigonometric Functions

Example 1

Evaluate .

Step 1.   Let u = 4x.

Step 2.   Differentiate: du = 4 dx or

Step 3.   Rewrite:

Step 4.   Integrate:

Step 5.   Replace u:

Step 6.   Differentiate and Check:

Example 2

Evaluate

Step 1.   Let u = tan x.

Step 2.   Differentiate: du = sec² x dx.

Step 3.   Rewrite:

Step 4.   Integrate:

Step 5.   Replace u: 2(tan x ) + c or 2 tan x + c

Step 6.   Differentiate and Check:

Example 3

Evaluate

Step 1.   Let u = x³.

Step 2.   Differentiate:

Step 3.   Rewrite:

Step 4.   Integrate:

Step 5.   Replace u:

Step 6.   Differentiate and Check:

U-Substitution and Inverse Trigonometric Functions

Example 1

Evaluate .

Step 1.   Let u =2x.

Step 2.   Differentiate: du =2x; .

Step 3.   .

Step 4.   .

Step 5.   .

Step 6.   .

Example 2

.

Step 1.   .

Let u = x +1.

Step 2.   Differentiate: du<1em> =dx.

Step 3.   .

Step 4.   .

Step 5.   .

Step 6.   .