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

Category: QuestionsFind \(\int_{π/4}^{π/2} \,2sinx \,sin⁡(cos⁡x) \,dx\).
Editor">Editor Staff asked 11 months ago

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
NCERT Solutions for Subject Clas 12 Math Select the correct answer from above options 
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1 Answers
Editor">Editor Staff answered 11 months ago

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