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Chapter 4: Problem 7

Find the integral. $$ \int \frac{1}{\sqrt{1-(x+1)^{2}}} d x $$

Short Answer

Expert verified
The integral \(\int \frac{1}{\sqrt{1-(x+1)^{2}}}\) dx is equal to arcsin(x+1) + C

Step by step solution

01

Identify the Integral Pattern

The first step is to identify the integral form. The given integral is a classic form of \(\int \frac{1}{\sqrt{1-u^{2}}}\) dx = arcsin(u) + C, where 'C' is the constant of integration.
02

Substitution

As per the above integral form, we can identify 'u' as '(x+1)'. So, we let u = x+1. Now the integral changes to \(\int \frac{1}{\sqrt{1-u^{2}}}\) du.
03

Apply the Integral

Applying the integral \(\int \frac{1}{\sqrt{1-u^{2}}}\) du = arcsin(u) + C, where 'C' is the constant of integration. So, the result is arcsin(u) + C.
04

Back-substitute for the Original Variable

Substituting u = x+1 back into the result gives the final answer, i.e., arcsin(x+1) + C.

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