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Q. Solve each of the integrals in Exercises 2770.Some integrals require integration by parts, and some do not.

x3cosh2xdx.

Short Answer

Expert verified

The value ofx3cosh2xdxis,4x3+6xsinh2x+-6x2-3cosh2x8+C.

Step by step solution

01

Step 1. Given information

x3cosh2xdx.

02

Step 2. Now evaluate the given integral.

Integrating by parts, fg'=fg-f'g.

Here, f=x3,g'=cosh2x.

f'=3x2,g=sinh2x2x3cosh2xdx=x3sinh2x2-3x2sinh2x2dx=x3sinh2x2-32x2sinh2xdx=x3sinh2x2-32x2cosh2x2-xcosh2xdx=x3sinh2x2-34x2cosh2x+32xcosh2xdx

03

Step 3. Use the Integration by parts method in the required places in the problem.

x3cosh2xdx=x3sinh2x2-34x2cosh2x+32xsinh2x2-sinh2x2dx=x3sinh2x2-34x2cosh2x+32xsinh2x2-32sinh2x2dx=x3sinh2x2-34x2cosh2x+32xsinh2x2-32cosh2x4=2x3sinh2x+3xsinh2x4+-6x2cosh2x-3cosh2x8=4x3+6xsinh2x8+-6x2-3cosh2x8=4x3+6xsinh2x+-6x2-3cosh2x8+C

Therefore, the required value is, 4x3+6xsinh2x+-6x2-3cosh2x8+C.

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