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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1 Answers
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