The u-Substitution Technique

Integration by substitution (also called u-substitution) is the integration counterpart of the chain rule for differentiation. It simplifies integrals by replacing a complicated expression with a single variable u.

If u = g(x), then du = g'(x) dx, and:
$\int f(g(x)) \cdot g'(x) dx = \int f(u) du$

1

Choose u: identify the "inner function" or the part that simplifies the integral.

2

Find du/dx and rearrange to express dx in terms of du.

3

Substitute u and du into the integral, replacing all x-terms.

4

Integrate in terms of u, then substitute back to get the answer in terms of x.

Indefinite Integrals with Substitution

Example: ∫ 2x(x² + 1)⁵ dx

Step 1: Let u = x² + 1, then du/dx = 2x, so du = 2x dx.

Step 2: The integral becomes ∫ u⁵ du.

Step 3: Integrate: u⁶/6 + C.

Step 4: Substitute back: (x² + 1)⁶/6 + C.

How to choose u: Look for a function and its derivative appearing in the integrand. The "inner function" in a composite expression is usually the best choice for u.

Changing Limits for Definite Integrals

For definite integrals, you have two options. The most efficient approach is to change the limits when substituting, so you never need to substitute back to x.

When x = a, u = g(a). When x = b, u = g(b).
$\int a b f(g(x))g'(x) dx = \int g(a) g(b) f(u) du$

Example: ∫₀¹ 2x(x² + 1)³ dx

Step 1: Let u = x² + 1, du = 2x dx.

Step 2: Change limits: when x = 0, u = 1; when x = 1, u = 2.

Step 3: ∫₁² u³ du = [u⁴/4]₁² = 16/4 − 1/4 = 15/4

Answer: 15/4

Key Vocabulary

u-Substitution

A technique that replaces a complex expression with a single variable u to simplify integration.

Substitution Back

Replacing u with the original expression in x after integrating (for indefinite integrals).

Changing Limits

Converting x-limits to u-limits for definite integrals, avoiding the need to substitute back.

Inner Function

The function inside a composite expression, typically the best choice for u.

Worked Examples

1

Find ∫ cos(3x) dx using substitution.

Step 1: Let u = 3x, then du = 3 dx, so dx = du/3.

Step 2: ∫ cos(u) · (du/3) = (1/3) ∫ cos(u) du = (1/3)sin(u) + C

Answer: (1/3)sin(3x) + C

2

Find ∫ x · e^x² dx.

Step 1: Let u = x², then du = 2x dx, so x dx = du/2.

Step 2: ∫ e^u · (du/2) = (1/2)e^u + C

Answer: (1/2)e^x² + C

3

Evaluate ∫₁² 3x²(x³ + 1)² dx using substitution.

Step 1: Let u = x³ + 1, du = 3x² dx.

Step 2: Change limits: x = 1 gives u = 2; x = 2 gives u = 9.

Step 3: ∫₂⁹ u⁻² du = [−u⁻¹]₂⁹ = −1/9 − (−1/2) = −1/9 + 1/2

Answer: 7/18

Knowledge Check

Select the correct answer for each question. Click "Check Answer" to see if you are right.

Question 1

To evaluate ∫ 6x(3x² + 5)⁴ dx, the best substitution is:

u = 6x u = x² u = 3x² + 5 u = (3x² + 5)⁴

Question 2

Find ∫ sin(2x) dx using substitution.

−cos(2x) + C −2cos(2x) + C (1/2)cos(2x) + C −(1/2)cos(2x) + C

Question 3

If u = x² + 3 and the x-limits are 0 to 1, what are the u-limits?

3 to 4 0 to 1 3 to 7 1 to 4

Question 4

Find ∫ 2xx² + 1 dx.

ln|2x| + C ln|x² + 1| + C (x² + 1)² + C −1/(x² + 1) + C

Question 5

Evaluate ∫₀^π/2 cos(x) · e^sin(x) dx.

e + 1 e e − 1 1

Integration by Substitution

The u-Substitution Technique

Indefinite Integrals with Substitution

Example: ∫ 2x(x² + 1)⁵ dx

Changing Limits for Definite Integrals

Example: ∫₀¹ 2x(x² + 1)³ dx

Key Vocabulary

u-Substitution

Substitution Back

Changing Limits

Inner Function

Worked Examples

Find ∫ cos(3x) dx using substitution.

Find ∫ x · e^x² dx.

Evaluate ∫₁² 3x²(x³ + 1)² dx using substitution.

Knowledge Check

Key Concepts Summary

Integration by Substitution

The u-Substitution Technique

Indefinite Integrals with Substitution

Example: ∫ 2x(x2 + 1)5 dx

Changing Limits for Definite Integrals

Example: ∫01 2x(x2 + 1)3 dx

Key Vocabulary

u-Substitution

Substitution Back

Changing Limits

Inner Function

Worked Examples

Find ∫ cos(3x) dx using substitution.

Find ∫ x · ex2 dx.

Evaluate ∫12 3x2(x3 + 1)2 dx using substitution.

Knowledge Check

Key Concepts Summary

Example: ∫ 2x(x² + 1)⁵ dx

Example: ∫₀¹ 2x(x² + 1)³ dx

Find ∫ x · e^x² dx.

Evaluate ∫₁² 3x²(x³ + 1)² dx using substitution.