Integration Using Standard Substitution

Integration using standard substitution, also known as u-substitution, is a technique used to simplify the process of finding antiderivatives. This method involves substituting a part of the integrand with a new variable to make the integral easier to solve.

When to Use Standard Substitution

Standard substitution is particularly useful when you encounter an integral that is not immediately integrable but can be transformed into an easier form. The goal is to rewrite the integral in terms of a new variable, u, which simplifies the integrand and makes it possible to integrate using basic integration rules.

How to Perform Standard Substitution

The process of standard substitution involves the following steps:

1. Identify a part of the integrand that can be substituted with a new variable u.
2. Differentiate u with respect to x to find du/dx.
3. Solve for dx in terms of du and u.
4. Substitute u and dx into the original integral.
5. Integrate with respect to u.
6. Substitute back the original variable x using the initial substitution.

Table of Common Substitutions

Here is a table of common substitutions and their corresponding differentials:

Substitution	Differential du	Condition
u = ax + b	du = a dx	a and b are constants
u = g(x)	du = g'(x) dx	g(x) is a differentiable function
u = sin(x)	du = cos(x) dx	-
u = cos(x)	du = -sin(x) dx	-
u = tan(x)	du = sec^2(x) dx	-
u = e^x	du = e^x dx	-
u = ln(x)	du = (1/x) dx	x > 0

Formulas for Integration by Substitution

After performing the substitution, you can use the following basic integral formulas:

• $\int du = u + C$
• $\int u^n du = \frac{u^{n+1}}{n+1} + C$, where n ≠ -1
• $\int e^u du = e^u + C$
• $\int \frac{1}{u} du = \ln|u| + C$

Examples

Example 1: Basic Substitution

Consider the integral $\int x \cdot e^{x^2} dx$. We can use the substitution $u = x^2$, which gives us $du = 2x dx$ or $\frac{1}{2} du = x dx$.

Substituting into the integral, we get:

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

Example 2: Trigonometric Substitution

For the integral $\int \sin(x) \cos(x) dx$, we can use the substitution $u = \sin(x)$, which gives us $du = \cos(x) dx$.

Substituting into the integral, we get:

$$ \int \sin(x) \cos(x) dx = \int u du = \frac{u^2}{2} + C = \frac{\sin^2(x)}{2} + C $$

Example 3: Logarithmic Substitution

Consider the integral $\int \frac{1}{x \ln(x)} dx$. We can use the substitution $u = \ln(x)$, which gives us $du = \frac{1}{x} dx$.

Substituting into the integral, we get:

$$ \int \frac{1}{x \ln(x)} dx = \int \frac{1}{u} du = \ln|u| + C = \ln|\ln(x)| + C $$

Important Points to Remember

• Always differentiate your substitution to find the correct differential du.
• Make sure the entire integrand is in terms of u before integrating.
• Don't forget to substitute back the original variable after integrating.
• Check your work by differentiating your answer to see if it matches the original integrand.

By mastering standard substitution, you can tackle a wide range of integrals that might initially seem challenging. Practice is key to becoming proficient in this technique.