1. Identify the Inner Function:

• Look for a function inside another function. This inner function is typically what you would have differentiated in the chain rule.

2. Substitute the Inner Function:

• Let $\ u = g(x)$, where $\ g(x)$ is the inner function.
• Differentiate $\ u$ with respect to $\ x$ to find $\ \frac{du}{dx}.$

3. Rewrite the Integral:

• Replace all occurrences of $\ g(x)$with $\ u \ and \ dx$ with $\ \frac{du}{\frac{du}{dx}} .$
• The integral should now be in terms of $\ u \ and \ du$, which may simplify the integration process.

4. Integrate with Respect to $\ u$:

• Perform the integration with respect to $\ u .$

5. Substitute Back:

• After finding the integral in terms of $u$, substitute $\ u = g(x)$ back into the expression to obtain the final answer in terms of $\ x$.

Example:

Evaluate $\ \int 2x \cdot \cos(x^2) \, dx .$

6. Identify the Inner Function:

• Here, $\ g(x) = x^2$, and the derivative $\ g'(x) = 2x .$

7. Substitute:

• Let $\ u = x^2 \ , so \ du = 2x \, dx .$

8. Rewrite the Integral:

• The integral becomes $\ \int \cos(u) \, du .$

9. Integrate:

• The integral of $\ \cos(u) \ is \ \sin(u) .$

10. Substitute Back:

• Replace $\ u \ with \ x^2$ to get the final answer: $\ \sin(x^2) + C .$

Example 1: Basic u-substitution

Question:

Evaluate $\int (3x^2)(x^3 + 1)^5 \, dx$

Solution:

11. Identify the inner function: Notice that $x^3 + 1$ is the inner function inside the power, and its derivative is $3x^2$, which is also in the integrand.

12. Let $u = x^3 + 1$: This gives:

13. Rewrite the integral in terms of $u$:

14. Integrate:

15. Substitute back $u = x^3 + 1:$

Thus, the solution is:

Example 2: Exponential Function

Question:

Evaluate $\int 2xe^{x^2} \, dx$.

Solution:

16. Identify the inner function: The function inside the exponential is $x^2$, and its derivative is $2x$, which is in the integrand.

17. Let $u = x^2$: This gives:

18. Rewrite the integral in terms of $u$:

19. Integrate:

20. Substitute back $u = x^2$:

Thus, the solution is:

Example 3: Trigonometric Function

Question:

Evaluate $\int \cos(3x) \, dx$.

Solution:

21. Identify the inner function: The inner function is $3x$, and its derivative is $3$.

22. Let $u = 3x$: This gives:

To match the integral, divide both sides by 3:

23. Rewrite the integral in terms of $u$:

24. Integrate:

25. Substitute back $u = 3x$:

Thus, the solution is:

Example 4: More Complex Polynomial

Question:

Evaluate $\int (2x + 1)^3 \cdot (x + 1) \, dx$

Solution:

26. Identify the inner function: The inner function $u = x + 1$ appears inside the cubic power.

27. Let $u = x + 1:$ This gives:

28. Rewrite the integral: Since $2x + 1 = 2(x + 1) - 1 = 2u - 1$ , the integral becomes:

29. Expand the expression: First expand $(2u - 1)^3$ :

Now multiply by $u$ :

30. Integrate each term:

Simplifying:

31. Substitute back $u = x + 1$: