Problem 7 - Essential Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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.

Step 3: 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.

Step 4: Back-substitute for the Original Variable

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