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 USubstitution 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 USubstitution
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 USubstitution
Steps:
 Given f(g(x)); let u = g(x).
 Differentiate: du = g '(x)dx
 Rewrite the integral in terms of u.
 Evaluate the integral.
 Replace u by g(x).
 Check your result by taking the derivative of the answer.
USubstitution and Algebraic Functions
Another Form of the Integral of a Composite Function
If f is a differentiable function, then
Making a USubstitution
Example 1
Evaluate x (x + 1)^{10}dx.
Step 1. Let u = x + 1; then x = u – 1.
Step 2. Differentiate: du = dx.
Step 3. Rewrite: (u – 1) u^{10}du = (u^{11} – u^{10})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)^{2/3}dx.
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 = x^{3} – 8.
Step 2. Differentiate: du =3x^{2}dx = x^{2}dx.
Step 3. Rewrite:
Step 4. Integrate:
Step 5. Replace
Step 6. Differentiate and Check:
USubstitution 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 = sec^{2} 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 = x^{3}.
Step 2. Differentiate:
Step 3. Rewrite:
Step 4. Integrate:
Step 5. Replace u:
Step 6. Differentiate and Check:
USubstitution 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. .

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