# 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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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

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

=2(1-cos⁡$$\frac{1}{\sqrt{2}}$$)