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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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} sint \,dt

=9[-cost]_0^{π/2}=-9(cos π/2-cos0)=-9(0-1)=9