Evaluate the integral \(\int_0^{\frac{π^2}{4}} \frac{9 sin⁡\sqrt{x}}{2\sqrt{x}} dx\).

Category: QuestionsEvaluate the integral \(\int_0^{\frac{π^2}{4}} \frac{9 sin⁡\sqrt{x}}{2\sqrt{x}} dx\).
Editor">Editor Staff asked 11 months ago

Evaluate the integral \int_0^{\frac{π^2}{4}} \frac{9 sin⁡\sqrt{x}}{2\sqrt{x}} dx.
(a) 9
(b) -9
(c) \frac{9}{2}
(d) –\frac{9}{2}
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The above asked question is from Evaluation of Definite Integrals by Substitution in section Integrals of Mathematics – Class 12
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1 Answers
Editor">Editor Staff answered 11 months ago

Right answer is (a) 9
The explanation: I=\int_0^{\frac{π^2}{4}} \frac{9 sin⁡\sqrt{x}}{2\sqrt{x}} dx
Let \sqrt{x}=t
Differentiating both sides w.r.t x, we get
\frac{1}{2\sqrt{x}} dx=dt
The new limits are
When x=0 , t=0
When x=\frac{π^2}{4}, t=\frac{π}{2}
∴\int_0^{\frac{π^2}{4}} \frac{9 sin⁡\sqrt{x}}{2\sqrt{x}} dx=9\int_0^{π/2} sin⁡t \,dt
=9[-cos⁡t]_0^{π/2}=-9(cos⁡ π/2-cos⁡0)=-9(0-1)=9