Integration by Parts Calculator

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Example

Created on 2024-06-20Asked by Victoria Adams (Solvelet student)

Evaluate the integral

\int x \cos x \, dx

using integration by parts.

Solution

To evaluate the integral

\int x \cos x \, dx

using integration by parts: Let

u = x

and

dv = \cos x \, dx

du = dx

v = \int \cos x \, dx = \sin x

Now, apply the integration by parts formula:

\int u \, dv = uv - \int v \, du

Substitute

u

du

, and

v

\int x \cos x \, dx = x \sin x - \int \sin x \, dx

Integrate

\sin x

\int \sin x \, dx = -\cos x

Therefore:

\int x \cos x \, dx = x \sin x + \cos x + C

Where

C

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

Some of the related questions asked by Evelyn Smith on Solvelet

1. Evaluate the integral

\int x \sin(x) \, dx

using integration by parts,2. Compute the integral

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

using integration by substitution.,

DefinitionIntegration by parts is a method for integrating the product of two functions. It relies largely on the product rule of differentiation and completes the integration of a product of functions to a more simple integrations. The formula is ∫udv=uv−∫vdu where u and dv are parts of the original integrand This type is invaluable when differentiation causes some parts of the product to become easier. Example: ∫xexdx du=x dv=exdx Then, du=dx and v=ex. Integration by parts to ∫xexdx=xex−∫exdx=xex−ex+C=ex(x−1)+C

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