Integration using substitution, often referred to as u-substitution, is a key technique in calculus for simplifying and evaluating complex integrals. This method involves changing the variable of integration to transform a difficult integral into a more manageable form. Mastering this technique is crucial for students preparing for their A-Level examinations in Pure Mathematics.

Understanding Substitution in Integration

Substitution in integration is like solving a puzzle, focusing on finding the right substitution to simplify the integral.

1. Identify Substitution: Choose a part of the integral as '$u$', ideally where its derivative appears in the integral.

2. Change Variables: Express '$dx$' in terms of '$du$', transforming the integral into a simpler form.

3. Integrate: Carry out the integration with the new variable '$u$'.

4. Revert Back: Replace '$u$' with the original variable to express the integral in original terms.

Examples of Integration Using Substitution

Example 1: Basic Substitution

Choose substitution: $u = x^2$, so $du = 2x dx$.

Modify the integral: becomes $\frac{1}{2} \int \sin(u) du$ after adjusting for $du$.

Integrate: $\sin(u)$ gives $-\frac{1}{2} \cos(u) + C$.

Return to original variable: Replace $u$ with $x^2$ for final answer: $-\frac{1}{2} \cos(x^2) + C$.

Example 2: Trigonometric Substitution

1. Choose substitution: $u = 2x$, hence $du = 2dx$.

2. Modify integral: becomes $\frac{1}{2} \int \sin^2(u) \cos\left(\frac{u}{2}\right) du$.

3. Simplify: Use trigonometric identities for $\sin^2(u)$ and $\cos\left(\frac{u}{2}\right)$.

4. Integrate: Find result as $\left(x - \frac{\sin(2x)\cos(2x)}{2}\right)\frac{\cos(x)}{2}$.

5. Return to original variable: Replace $u$ with $2x$.

Integration of Special Functions

Some functions require specific substitutions to simplify the integration process. For instance, integrals involving $\sqrt{a^2 - x^2}$ often benefit from trigonometric substitutions like $x = a \sin(u)$.

Example:

Integrating $\int \frac{1}{\sqrt{1 - x^2}} \, dx$ Using Trigonometric Substitution:

1. Set $x = \sin(u)$, then $dx = \cos(u) du$.

2. Transform integral to $\int \frac{\cos(u)}{\sqrt{1 - \sin^2(u)}} du$.

3. Simplify using $\sin^2(u) + \cos^2(u) = 1$, getting $\int 1 du$.

4. Integrate to find $u$.

5. Revert variable,$x = \sin(u)$ implies $u = \sin^{-1}(x)$.

6. Final result: $\sin^{-1}(x) + C$.