   Chapter 8.2, Problem 44E

Chapter
Section
Textbook Problem

# Evaluating a Definite Integral In Exercises 43-52, evaluate the definite integral. Use a graphing utility to verify your result. ∫ 0 1 x arcsin x 2   d x

To determine

To calculate: The find the value of definite integral 01xarcsinx2dx.and also verify it by using graphing utility.

Explanation

Given: the definite integral is given as 01xarcsinx2dx

Formula used:

By using Integral by part method, i.e.

udv=uvvdu

Explanation:

The given integration we have,

01xarcsinx2dx

To Find integration by part method

Let u=arcsinx2

On Differentiating the function with respect to x

We have,

ddxu=ddx(sin1(x2))dudy=2x1x4

And

dv=xdx

Now, on integrating both sides

we get,

v=x22

by using above mentioned formula of integral by part method ,

we have

udv=uvvdu

On putting values above

We get,

xarcsinx2dx=x22sin1x22x1x4×x22dx=x22sin1x2x31x4dx …… (1)

Now, on Considering, x31x2dx

Let t=1x4

On Differentiating the function on both sides with respect to x.

We get,

dt=4x3dxdt4=x3dx

On Substituting the values in the integration,

x31x2dx=1t(dt4)=14(t)12+1×2=12t

Here, put t=1x4 then

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