Integration by Substitution Calculator

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Example

Created on 2024-06-20Asked by Emma Green (Solvelet student)

Evaluate the integral

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

using substitution.

Solution

To evaluate the integral

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

using substitution: Let

u = x^3 + x

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

The integral becomes:

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

Integrate

u

\int u \, du = \frac{u^2}{2} + C

Substitute back

u = x^3 + x

\frac{(x^3 + x)^2}{2} + C

Therefore, the integral

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

is:

\int (3x^2 + 1)(x^3 + x) \, dx = \frac{(x^3 + x)^2}{2} + C

Where

C

is the constant of integration. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Mateo Walker on Solvelet

1. Evaluate the integral

\int (2x + 3)\sqrt{4x^2 + 6x + 5} \, dx

using integration by substitution,2. Compute the area enclosed by the curve

y = 2\sqrt{1 - x^2}

and the

x

-axis using trigonometric substitution.,

DefinitionNow, this is a simple case of variable substitution or u-substitution or simply, Integration by substitution. This is the method of replacement, and it consists in replacing the integral by one written with u as the variable and then the integral. The substitution typically involves recognizing a function and its derivative in the integrand. For instance, if you were evaluating the following integral: ∫xx2+1dx your u value would be u=x2+1. Then, du=2xdx, so xdx=21du. The integral simplifies to 12 ∫u du = 21∫u1/2du. Therefore integrating we get 21⋅32u3/2+C=31(x2+1)3/2+C

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