# Find $$\int_0^{\frac{\sqrt{π}}{2}} 2x \,cos⁡ x^2 \,dx$$.

Category: QuestionsFind $$\int_0^{\frac{\sqrt{π}}{2}} 2x \,cos⁡ x^2 \,dx$$.
Editor">Editor Staff asked 11 months ago

Find $$\int_0^{\frac{\sqrt{π}}{2}} 2x \,cos⁡ x^2 \,dx$$.

(a) 1

(b) $$\frac{1}{\sqrt{2}}$$

(c) –$$\frac{1}{\sqrt{2}}$$

(d) $$\sqrt{2}$$

This question was posed to me in unit test.

This interesting 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

Right choice is (b) $$\frac{1}{\sqrt{2}}$$

The explanation is: I=$$\int_0^{\frac{\sqrt{π}}{2}} \,2x \,cos⁡ x^2 \,dx$$

Let x^2=t

Differentiating w.r.t x, we get

2x dx=dt

The new limits

When x=0,t=0

When $$x={\frac{\sqrt{π}}{2}}, t=\frac{π}{4}$$

∴$$\int_0^{\frac{\sqrt{π}}{2}} \,2x \,cos⁡ x^2 \,dx=\int_0^{\frac{π}{4}} \,cos⁡t \,dt$$

$$I =[sin⁡t]_0^{\frac{π}{4}}=sin⁡ \frac{π}{4}-sin⁡0=1/\sqrt{2}$$.