Integration Techniques Calculator

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Example

Created on 2024-06-20Asked by Benjamin Thomas (Solvelet student)

Evaluate the integral

\int e^{3x} \cos x \, dx

using an appropriate integration technique.

Solution

To evaluate the integral

\int e^{3x} \cos x \, dx

: We will use integration by parts. Let

u = e^{3x}

and

dv = \cos x \, dx

du = 3e^{3x} \, 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 e^{3x} \cos x \, dx = e^{3x} \sin x - \int \sin x \cdot 3e^{3x} \, dx

Simplify:

= e^{3x} \sin x - 3 \int e^{3x} \sin x \, dx

Now, use integration by parts again on

\int e^{3x} \sin x \, dx

: Let

u = e^{3x}

and

dv = \sin x \, dx

du = 3e^{3x} \, dx

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

Substitute:

\int e^{3x} \sin x \, dx = -e^{3x} \cos x + \int \cos x \cdot 3e^{3x} \, dx

Substitute back into the original equation:

\int e^{3x} \cos x \, dx = e^{3x} \sin x - 3 \left( -e^{3x} \cos x + \int e^{3x} \cos x \, dx \right)

Simplify and solve for the integral:

\int e^{3x} \cos x \, dx = e^{3x} \sin x + 3e^{3x} \cos x - 3 \int e^{3x} \cos x \, dx

4 \int e^{3x} \cos x \, dx = e^{3x} (\sin x + 3 \cos x)

\int e^{3x} \cos x \, dx = \frac{e^{3x} (\sin x + 3 \cos x)}{4} + C

Therefore, the integral

\int e^{3x} \cos x \, dx

is:

\int e^{3x} \cos x \, dx = \frac{e^{3x} (\sin x + 3 \cos x)}{4} + C

Where

C

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

Some of the related questions asked by Aria Lee on Solvelet

1. Evaluate the integral

\int \frac{x^2 + 3x + 2}{x + 1} \, dx

using partial fraction decomposition,2. Use polynomial interpolation to estimate the value of

y

when

x = 2.5

based on the data points

(1, 2)

(2, 4)

, and

(3, 8)

DefinitionMethods and strategies to find the integrals of functions are called the Integration techniques. Substitution, integration by parts, along with more complex methods like partial fractions, trig substitution, and complex contour integration are among the tools available. Each of these techniques is appropriate for different kinds of integrands, and can really help in reducing the integration to a simpler form. Description: Standard tools include substitution for integrals of functions of functions, and integration by parts for products of functions, such as the integral of x times e to the x. For rational functions etc we use more advanced techniques like partial fractions, e.g. ∫(x−1)(x+2)1dx

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