   Chapter 7.R, Problem 34E

Chapter
Section
Textbook Problem

# Evaluate the integral. ∫ ( arcsin x ) 2 d x

To determine

To Find:(arcsinx)2dx

Explanation

Formula Used:uvdt=uvdtu'(vdt)dt

Calculation:

Here, the trigonometric substitution will be helpful for the kind of integrand given.

Use the substitution x=sint. Now, differentiate x=sint with respect to x to get

dx=costdt

After substitution, the integral reduces to

(arcsinx)2dx=t2costdt

Now, we have the integrand as the product of two functions so it is advisable to use integration by parts or in other words use the formula mentioned above by taking u=t2 and v=cost o get

(arcsinx)2dx=t2costdt=t2costdtddt(t2)(costdt)dt=t2sinx2tsintdt

Again, use the integration by parts to find the remaining integral

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