Harder Substitution (AQA A-Level Mathematics): Revision Notes

📑Example

Problem

Integrate the following function with respect to $x$ :

Step-by-Step Solution

1. Identify a suitable substitution: We notice that the expression $1 + x^2$ is inside the square root, and its derivative $2x$ is almost present in the integrand (we have $x$ ). This suggests that a good substitution could be:

$u = 1 + x^2$

2. Differentiate $\ u$ with respect to $\ x :$

$\frac{du}{dx} = 2x$

Therefore, we can express $dx$ in terms of $du$:

$dx = \frac{du}{2x}$

3. Rewrite the integral in terms of $u$: Substitute $u = 1 + x^2$ and $dx = \frac{du}{2x}$ into the original integral:

$\int x \sqrt{1 + x^2} \, dx = \int x \sqrt{u} \cdot \frac{du}{2x}$

Notice that the $x$ terms cancel out:

$\int x \sqrt{u} \cdot \frac{du}{2x} = \frac{1}{2} \int \sqrt{u} \, du$

4. Integrate with respect to $u$ : The integral now simplifies to:

$\frac{1}{2} \int u^{\frac{1}{2}} \, du$

Use the power rule for integration:

$\frac{1}{2} \cdot \frac{u^{\frac{3}{2}}}{\frac{3}{2}} = \frac{1}{2} \cdot \frac{2}{3} u^{\frac{3}{2}} = \frac{1}{3} u^{\frac{3}{2}}$

5. Substitute back $u = 1 + x^2$: Finally, replace $u$ with the original expression in terms of $x$ :

$\frac{1}{3} (1 + x^2)^{\frac{3}{2}} + C$

Where $C$ is the constant of integration.

Final Answer

$\int x \sqrt{1 + x^2} \, dx = \frac{1}{3} (1 + x^2)^{\frac{3}{2}} + C$

If you haven't been doing so already, make sure you attempt the example questions!

Example 1: Involving Trigonometric Functions

Question:

Evaluate $\int \frac{\sin(2x)}{1 + \cos(2x)} \, dx$.

Solution:

6. Recognize the structure of the integrand: The expression $1 + \cos(2x)$ suggests the use of a trigonometric identity. Use the identity:

But here, using substitution directly is more straightforward.

7. Let $u = 1 + \cos(2x)$: This gives:

We need $\sin(2x) \, dx$ , so divide by $-2$:

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

9. Simplify and integrate:

10. Substitute back $u = 1 + \cos(2x)$:

Thus, the solution is:

Example 2: Rational Functions

Question:

Evaluate $\int \frac{1}{x^2 \sqrt{x^2 - 1}} \, dx$.

Solution:

11. Recognize the structure of the integrand: The term $\sqrt{x^2 - 1}$ suggests a trigonometric substitution. We know that $\sec^2(\theta) - 1 = \tan^2(\theta)$ .

12. Substitute $x = \sec(\theta)$: This gives:

13. Rewrite the integral:

Substituting into the integral:

14. Simplify:

15. Integrate:

16. Substitute back $\theta = \sec^{-1}(x)$ : Since $\sin(\theta) = \frac{\sqrt{x^2 - 1}}{x}$ , the final solution is:

Example 3: Exponential and Polynomial Mix

Question:

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

Solution:

17. Recognize the structure of the integrand: The term $x^2$ is in the exponent, and its derivative is $2x$, which suggests substitution.

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

We need $x \, dx$, so divide both sides by 2:

19. Rewrite the integral:

20. Integrate:

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

Thus, the solution is:

Example 4: Involving Logarithms

Question:

Evaluate $\int \frac{\ln(x)}{x} \, dx$.

Solution:

22. Recognize the structure of the integrand: The function $\ln(x)$ appears in the numerator, and its derivative is $\frac{1}{x}$, suggesting substitution.

23. Let $u = \ln(x)$: This gives:

24. Rewrite the integral:

25. Integrate:

26. Substitute back $u = \ln(x)$:

Thus, the solution is:

Example 5: Multiple Substitutions

Question:

Evaluate $\int \frac{dx}{(1 + x^2)^{3/2}}$.

Solution:

27. Recognize the structure of the integrand: The term $1 + x^2$ suggests a trigonometric substitution since $\sec^2(\theta)$or $\tan^2(\theta)$often appear with such expressions.

28. Substitute $x = \tan(\theta)$: This gives:

Also, $1 + x^2 = 1 + \tan^2(\theta) = \sec^2(\theta)$.

29. Rewrite the integral:

30. Integrate:

31. Substitute back $\theta = \tan^{-1}(x)$:

Thus, the solution is: