Integration by Substitution

Integration by Substitution

Integration by substitution (also called u-substitution) is a technique used to simplify integrals by transforming them into a more manageable form. It is similar to the chain rule in differentiation but applied in reverse.

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1. The Substitution Rule

If an integral is of the form:

$$\int f(g(x)) g'(x) \, dx$$

we use the substitution:

$$u = g(x), \quad \text{so that} \quad du = g'(x) \, dx$$

This transforms the integral into:

$$\int f(u) \, du$$

which is often easier to evaluate.

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2. Steps for Integration by Substitution

1. Choose a substitution: Let $u = g(x)$, where $g(x)$ is a function inside the integral.
2. Differentiate $u$: Compute $du = g'(x) dx$.
3. Rewrite the integral: Express everything in terms of $u$, replacing $dx$ with $du$.
4. Evaluate the new integral: Solve the integral in terms of $u$.
5. Substitute back $x$: Replace $u$ with $g(x)$ to return to the original variable.

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3. Examples of Integration by Substitution

Example 1: $\int 2x (x^2 + 1)^3 \, dx$

Step 1: Choose a substitution
Let:

$$u = x^2 + 1$$

Step 2: Differentiate

$$du = 2x \, dx$$

Step 3: Rewrite the integral

$$\int 2x (x^2 + 1)^3 \, dx = \int u^3 \, du$$

Step 4: Integrate

$$\frac{u^4}{4} + C$$

Step 5: Substitute back $u = x^2 + 1$

$$\frac{(x^2 + 1)^4}{4} + C$$

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Example 2: $\int \frac{\sin x}{\cos x} \, dx$

Step 1: Choose a substitution
Let:

$$u = \cos x$$

Step 2: Differentiate

$$du = -\sin x \, dx$$

Step 3: Rewrite the integral

$$\int \frac{\sin x}{\cos x} \, dx = -\int \frac{du}{u}$$

Step 4: Integrate

$$-\ln |u| + C$$

Step 5: Substitute back $u = \cos x$

$$-\ln |\cos x| + C$$

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Example 3: Definite Integral $\int_{0}^{1} x e^{x^2} \, dx$

Step 1: Choose a substitution
Let:

$$u = x^2$$

Step 2: Differentiate

$$du = 2x \, dx \quad \Rightarrow \quad \frac{du}{2} = x \, dx$$

Step 3: Rewrite the integral

$$\int_{0}^{1} x e^{x^2} \, dx = \frac{1}{2} \int_{0}^{1} e^u \, du$$

Step 4: Evaluate the integral

$$\frac{1}{2} e^u \Big|_{0}^{1}$$ $$= \frac{1}{2} (e^1 - e^0)$$ $$= \frac{1}{2} (e - 1)$$

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4. When to Use Substitution?

• When you see a composite function $f(g(x))$.
• If the integral contains a function and its derivative (or a multiple of it).
• When standard integration rules do not apply directly.

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5. Applications of Integration by Substitution

• Physics: Solving motion equations with changing acceleration.
• Engineering: Finding energy and work done by varying forces.
• Probability: Computing probability distributions.
• Economics: Finding marginal changes in cost and revenue.

Integration by substitution is one of the most important techniques for solving integrals, helping simplify otherwise complex expressions.