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Solve given definite integral.

35x29dx

Short Answer

Expert verified

92cosh153+94sinh2cosh153

Step by step solution

01

Step1. Given Information

The integral is as follows.

35x29dx

The objective is to solve the integral.

02

Step2. Assumptions

To solve the integral, let x=3cosh(u)

Now, derivative of above equation is solved below.

x=3cosh(u)dx=3sinh(u)du

03

Step3. Solution

Now, use the identity cosh2(u)1=sinh2(u).

So,

x29=9cosh2(u)9=3cosh2(u)1=3sinh2(u)=3sinh(u)

04

Step4. Solution

The integral is solved below.

9sinh2(u)du=9sinh2(u)du=912cosh(2u)12du=912du+92cosh(2u)du=92u+94sinh(2u)=92cosh1x3+94sinh2cosh1x3

05

Step5. Solution

The definite integral is solved below.

35x29dx=94sinh2cosh1x392cosh1x335=94sinh2cosh15392cosh15394sinh2cosh13392cosh133=92cosh153+94sinh2cosh153

Therefore, the value is92cosh153+94sinh2cosh153

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