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Question

Mathematics Question on Integrals of Some Particular Functions

∫18+2x−x2dx\int \frac{1}{\sqrt{8+2x-x^{2}}} dx is equal to

A

13sin−1(x−13)+c\frac{1}{3} sin^{-1} \left(\frac{x-1}{3}\right) +c

B

sin−1(x+13)+csin^{-1} \left(\frac{x+1}{3}\right) +c

C

13sin−1(x+13)+c\frac{1}{3}sin^{-1} \left(\frac{x+1}{3}\right) +c

D

sin−1(x−13)+csin^{-1} \left(\frac{x-1}{3}\right) +c

Answer

sin−1(x−13)+csin^{-1} \left(\frac{x-1}{3}\right) +c

Explanation

Solution

∫dx8+2x−x2=∫dx8−(x2−2x+1)+1\int \frac{dx}{\sqrt{8+2x-x^{2}}}=\int \frac{dx}{\sqrt{8-\left(x^{2}-2x+1\right)+1}}
=∫dx32−(x−1)2=sin⁡−1(x−13)+c=\int \frac{dx}{\sqrt{3^{2}-(x-1)^{2}}}=\sin ^{-1}\left(\frac{x-1}{3}\right)+c