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Prove the integration formula.

tan-1xdx=xtan-1x-12lnx2+1+C

(a) by applying integration by parts to tan-1xdx.

(a) by differentiatingxtan-1x-12lnx2+1.

Short Answer

Expert verified

Part (a). The solution is xtan-1x-12lnx2+1+C.

Part (b). The solution istan-1x.

Step by step solution

01

Part (a) Step 1. Given information.

The given integral is:

tan-1xdx=xtan-1x-12lnx2+1+C

02

Part (a) Step 2. Prove the formula by applying integration by parts.

Take u=tan-1xand v=1.

tan-1x·1dx=tan-1xdx-ddxtan-1xdxdx=x·tan-1x-x1+x2dx=x·tan-1x-121tdtput1+x2=dt,2xdx=dt,xdx=12dt=x·tan-1x-12ln(t)+C=x·tan-1x-12lnx2+1+C

Hence, the formula is proved.

03

Part (b) Step 1. Given information.

The given formula istan-1xdx=x·tan-1x-12lnx2+1+C.

04

Part (b) Step 2. Prove the formula by differentiating x·tan-1x-12lnx2+1.

ddxx·tan-1x-12lnx2+1=xddxtan-1x+tan-1xddxx-12ddxlnx2+1=x1+x2+tan-1x-12·11+x2·2xapplychainruleofdifferentiation=x1+x2+tan-1x-x1+x2=tan-1x

Hence, the formula is proved.

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