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Solvex21+x2dx the following two ways:

(a) with the substitution u=tan-1x;

(b) with the trigonometric substitution x = tan u.

Short Answer

Expert verified

Part (a) The solution of the given integral is x-tan-1x+C.

Part (b) The solution of the given integral is x-tan-1x+C.

Step by step solution

01

Part (a) Step 1. Given Information.

The given integral isx21+x2dx.

02

Part (a) Step 2. Solve.

We have to solve the integral with the substitution u=tan-1x,so the derivative is du=11+x2dx.

Let's solve the integral by substituting u,

x21+x2dx=tanu21+tanu21+tanu2du=tan2udu=-1+sec2udu=-1du+sec2udu=-u+tanu+CSubstitutebacku=-tan-1x+x+C=x-tan-1x+C

03

Part (b) Step 1. Solve.

We have to solve the integral with the substitution x=tanu,so the derivative is dx=sec2udu.

Let's solve the integral by substituting x,

role="math" localid="1648810032401" x21+x2dx=tan2u1+tan2usec2udu=tan2usec2usec2udu1+tan2u=sec2u=tan2udu=-1+sec2udu=-1du+sec2udu=-u+tanu+CSubstitutebacku=-tan-1x+x+C=x-tan-1x+C

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