# 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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Editor">Editor Staff answered 11 months ago

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