Question

Find \(\int_{π/4}^{π/2} \,2sinx \,sin⁡(cos⁡x) \,dx\).

(a) 2(1-cos⁡\(\frac{1}{\sqrt{2}}\))

(b) (cos⁡\(\frac{1}{\sqrt{2}}\)-cos⁡1)

(c) 2(cos⁡\(\frac{1}{\sqrt{2}}\)+1)

(d) (cos⁡\(\frac{1}{\sqrt{2}}\)+cos⁡1)

This question was posed to me in semester exam.

Question is taken from Fundamental Theorem of Calculus-1 in chapter Integrals of Mathematics – Class 12

Answer 1

Right option is (a) 2(1-cos⁡\(\frac{1}{\sqrt{2}}\))

For explanation I would say: Let \(I=\int_{π/4}^{π/2} \,2sinx \,sin⁡(cos⁡x) \,dx\)

F(x)=\(\int 2 \,sin⁡x \,sin⁡(cos⁡x)dx\)

Let cos⁡x=t

Differentiating w.r.t x, we get

sin⁡x dx=dt

∴\(\int 2 \,sin⁡x \,sin⁡(cos⁡x)dx=\int 2 \,sin⁡t \,dt=-2 \,cos⁡t\)

Replacing t with cos⁡x, we get

∴∫ 2 sin⁡x sin⁡(cos⁡x)dx=-2 cos⁡(cos⁡x)

By applying the limits, we get

\(I=F(\frac{π}{4})-F(\frac{π}{2})=-2 cos⁡(\frac{cos⁡π}{4})+2 cos⁡(\frac{cos⁡π}{2})\)

=2(1-cos⁡\(\frac{1}{\sqrt{2}}\))