Integration Using Substitution Calculator

This calculator helps you solve definite and indefinite integrals using the substitution method (also called u-substitution). Learn how to perform substitution integration, understand the substitution rule, and see worked examples.

What is substitution in integration?

Substitution integration is a technique used to simplify integrals that contain composite functions. The substitution method allows you to transform a complex integral into a simpler one by making a substitution for part of the integrand.

The substitution rule is based on the chain rule from differentiation. If you know the derivative of a function, you can use it to find the antiderivative through substitution.

Substitution integration is particularly useful when dealing with integrals that contain nested functions, such as ∫x²e^(x³) dx or ∫sin(x)/cos(x) dx.

How to use the substitution method

Step 1: Identify the substitution

Choose a substitution u that represents the inner function of the integrand. For example, in ∫x²e^(x³) dx, you might let u = x³ because its derivative is 3x², which appears in the integrand.

Step 2: Find the derivative

Differentiate your substitution to find du/dx. In the example above, du/dx = 3x², so du = 3x² dx.

Step 3: Rewrite the integral

Express the original integral in terms of u. In our example, we can write dx = du/3x², so the integral becomes ∫e^u (du/3x²).

Step 4: Integrate with respect to u

Now you can integrate with respect to u. The integral of e^u is e^u, so the result is e^u/3x² + C.

Step 5: Substitute back

Replace u with the original expression to get the final answer: e^(x³)/3x² + C.

The substitution rule can be written as:

∫f(g(x))g'(x) dx = f(g(x)) + C

Worked examples

Example 1: ∫x e^(x²) dx

Let u = x², du = 2x dx
Rewrite the integral: (1/2)∫e^u du
Integrate: (1/2)e^u + C
Substitute back: (1/2)e^(x²) + C

Example 2: ∫sin(x)/cos(x) dx

Let u = cos(x), du = -sin(x) dx
Rewrite the integral: -∫du/u
Integrate: -ln|u| + C
Substitute back: -ln|cos(x)| + C

Comparison of substitution results
Integral	Substitution	Result
∫x e^(x²) dx	u = x²	(1/2)e^(x²) + C
∫sin(x)/cos(x) dx	u = cos(x)	-ln\|cos(x)\| + C

Substitution formula

The substitution rule for indefinite integrals is:

∫f(g(x))g'(x) dx = f(g(x)) + C

For definite integrals, the limits must be adjusted accordingly:

∫[a,b] f(g(x))g'(x) dx = f(g(b)) - f(g(a))

The substitution method is valid when the antiderivative of the outer function exists and when the substitution is one-to-one.

FAQ

When should I use substitution integration?: Use substitution when the integrand contains a composite function and you can identify a substitution that simplifies the integral.
What if my substitution doesn't simplify the integral?: If your substitution doesn't simplify the integral, try a different substitution or consider other integration techniques like integration by parts or trigonometric identities.
Can substitution be used for definite integrals?: Yes, substitution can be used for definite integrals. You'll need to adjust the limits of integration according to your substitution.
What if the derivative of my substitution doesn't appear in the integrand?: If the derivative of your substitution doesn't appear in the integrand, you may need to include a coefficient when rewriting the integral.
Is substitution always the best method for integration?: No, substitution is just one of several integration techniques. Choose the method that best simplifies your particular integral.