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NTA Abhyas 2020
Exam
NTA Abhyas
Year
2020
Subject
Mathematics
Chapter
Indefinite Integration

NTA Abhyas 2020 · Mathematics · Indefinite Integration

If f(x) = sin x, g(x) = cos x and h(x) = (cos x), then the integral...

If f(x)=sinx,g(x)=cosxf\left(x\right) = sin x, g\left(x\right) = cos x and h(x)=cos⁡(cosx),h\left(x\right) = \cos \left(cos x\right), then the integral I=∫fΙ = ∫ f\left(g\left(x\right)\right) . f\left(x\right) . h\left(x\right)dx simplifies to −λsin⁡2(cosx)+C- \lambda\sin^{2} \left(cos x\right) + C (where, CC is the constant of integration). The value of λ\lambda is equal to

Correct answer

0.50

Step-by-step solution

Given integral, I=∫sin⁡(cosx).sinx.cos⁡(cosx)dxΙ = ∫ \sin \left(cos x\right) . sin x . \cos \left(cos x\right)dx
Let, sin⁡(cosx)=t⇒−cos⁡(cosx).sinxdx=dt\sin \left(cos x\right) = t ⇒ - \cos \left(cos x\right) . sin xdx = dt
∴I=∫−tdt=−t22+C∴ Ι = ∫ - tdt = - \dfrac{{t}^{2}}{2} + C
=C−sin2(cosx)2= C - \dfrac{si{n}^{2}\left(cos x\right)}{2}
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