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

(a) 2(1-cos\(\frac{1}{\sqrt{2}}\))

(b) (cos\(\frac{1}{\sqrt{2}}\)-cos1)

(c) 2(cos\(\frac{1}{\sqrt{2}}\)+1)

(d) (cos\(\frac{1}{\sqrt{2}}\)+cos1)

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

NCERT Solutions for Subject Clas 12 Math Select the correct answer from above options

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

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

F(x)=\(\int 2 \,sinx \,sin(cosx)dx\)

Let cosx=t

Differentiating w.r.t x, we get

sinx dx=dt

∴\(\int 2 \,sinx \,sin(cosx)dx=\int 2 \,sint \,dt=-2 \,cost\)

Replacing t with cosx, we get

∴∫ 2 sinx sin(cosx)dx=-2 cos(cosx)

By applying the limits, we get

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

=2(1-cos\(\frac{1}{\sqrt{2}}\))