Question

Find \(\int \frac{6 sin⁡\sqrt{x}}{\sqrt{x}} dx\)

(a) \(2 \,cos⁡\sqrt{x}+C\)

(b) –\(12 \,cos⁡\sqrt{x}+C\)

(c) -12 cos⁡x+C

(d) 12 cos⁡x+C

This question was addressed to me at a job interview.

This interesting question is from Methods of Integration-1 topic in division Integrals of Mathematics – Class 12

Answer 1

Correct answer is (b) –\(12 \,cos⁡\sqrt{x}+C\)

Easiest explanation: Let \(\sqrt{x}=t\)

Differentiating w.r.t x,we get

\(\frac{1}{2\sqrt{x}} dx=dt\)

\(\frac{1}{\sqrt{x}} dx=2dt\)

∴\(\int \frac{6 sin⁡\sqrt{x}}{\sqrt{x}} dx=\int \,12 \,sin⁡t \,dt\)

=12(-cos⁡t)=-12 cos⁡t

Replacing t with \(\sqrt{x}\), we get

\(\int \frac{6 sin⁡\sqrt{x}}{\sqrt{x}} dx=-12 \,cos⁡\sqrt{x}+C\)