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}
 
This question was posed to me in examination.
 
The above asked question is from Evaluation of Definite Integrals by Substitution in section Integrals of Mathematics – Class 12
NCERT Solutions for Subject Clas 12 Math Select the correct answer from above options 
Interview Questions and Answers, Database Interview Questions and Answers for Freshers and Experience

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