Question

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

Answer 1

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