# Find $$\int \frac{6 sin⁡\sqrt{x}}{\sqrt{x}} dx$$

Category: QuestionsFind $$\int \frac{6 sin⁡\sqrt{x}}{\sqrt{x}} dx$$
Editor">Editor Staff asked 11 months ago

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

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This interesting question is from Methods of Integration-1 topic in division Integrals of Mathematics – Class 12
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Editor">Editor Staff answered 11 months ago

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