integration by substitution

Integration by substitution (also called u-substitution) is one of the most powerful tools in calculus. You can think of it as the **reverse of the derivatives chain rule ** for differentiation.

Its goal is to simplify a messy, complex integral by changing the variable from $x$ to a new variable, usually $u$.

The Core Concept

The method works best when your integral contains a composite function $f(g(x))$ multiplied by the derivative of the “inside” function $g^{\prime }(x)$.

The formal formula is:

$$\int f(g(x))\cdot g^{\prime }(x)dx=\int f(u)du$$

Where:

• $u=g(x)$ (The “inside” function)
• $du=g^{\prime }(x)dx$ (The derivative of the inside function)

---

The Step-by-Step Method

Here is the standard workflow to solve an integral using substitution:

1. Choose $u$: Look for a function inside another function (e.g., inside a power, a square root, a sine, or an exponent). Ideally, the derivative of $u$ should also appear somewhere in the integrand.
2. Differentiate: Find the differential $du$ by taking the derivative of your chosen $u$. $du=\frac{du}{dx}\cdot dx$
3. Isolate $dx$: Solve for $dx$ so you can replace it in the original integral.
4. Substitute: Rewrite the entire integral in terms of $u$. All $x$‘s must disappear.
5. Integrate: Solve the new, simpler integral with respect to $u$.
6. Back-Substitute: If it is an indefinite integral, replace $u$ back with the original expression in terms of $x$.

---

Example 1: Indefinite Integral

Problem: Calculate $\int 2x(x^2+1{)}^{4}dx$

1. Choose $u$: The “inside” function is usually the complex part inside the parentheses. Let $u=x^2+1$.

2. Differentiate: $\frac{du}{dx}=2x⟹du=2xdx$

3. Substitute: Notice that $2xdx$ is exactly what we have in the integral. We can replace $(x^2+1)$ with $u$ and $(2xdx)$ with $du$. $\int (x^2+1{)}^4\cdot 2xdx=\int u^{4}du$

4. Integrate: $\int u^{4}du=\frac{u^5}{5}+C$

5. Back-Substitute: Replace $u$ with $(x^2+1)$. $\text{Answer: }\frac{(x^2+1{)}^5}{5}+C$

---

Example 2: Definite Integral (Changing Bounds)

When dealing with definite integrals (integrals with limits), you must change the limits of integration from $x$-values to $u$-values.

Problem: Calculate ${\int }_0^{1}x(x^2+1{)}^{3}dx$

1. Choose $u$: Let $u=x^2+1$.

2. Find $du$: $du=2xdx⟹dx=\frac{du}{2x}$

3. Change Limits:

• Lower limit: When $x=0$, $u=0^2+1=1$.
• Upper limit: When $x=1$, $u=1^2+1=2$.

4. Substitute: ${\int }_1^{2}x(u{)}^3\cdot \frac{du}{2x}$ The $x$‘s cancel out: $\frac{1}{2}{\int }_1^2{u}^{3}du$

5. Integrate and Evaluate: (Note: You do not back-substitute to $x$ because we changed the bounds to $u$). $\frac{1}{2}{\left[\frac{u^4}{4}\right]}_1^2$ $=\frac{1}{8}(2^4-1^4)$ $=\frac{1}{8}(16-1)=\frac{15}{8}$

---

When to use Substitution

Look for these patterns:

• Function and its Derivative: e.g., $\int \cos (x)\sin (x)dx$ (Derivative of sine is cosine).
• Composite Functions: Expressions like $e^{x^2}$, $\sqrt{5x+2}$, or $\frac{1}{3x-4}$.
• Logarithmic Forms: Integrals that look like $\int \frac{g^{\prime }(x)}{g(x)}dx$, which integrate to $\ln |g(x)|$.