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
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1 Answers
Editor">Editor Staff answered 11 months ago

Right answer is (a) 9
 
The explanation: I=

π
2
4
0
9sin
x



2
x



dx
 
 
Let
x



=t
 
Differentiating both sides w.r.t x, we get
 
1
2
x



dx=dt
 
 
The new limits are
 
When x=0 , t=0
 
When x=
π
2
4
,t=
π
2
 
 


π
2
4
0
9sin
x



2
x



dx=9

π/2
0
sintdt
 
 
=
9[−cost
]
π/2
0
=-9(cos⁡ π/2-cos⁡0)=-9(0-1)=9