If f(x)=sin x,g(x)=cos⁡x and h(x)=cos(cos x), then the integral I =∫ f(g(x)) ⋅ f(x) ⋅ h(x) d x simplifies to -λ (sin)2 (cos ⁡ x)+C (where, C is the constant of integration). The value of λ is equal to

Question

If f(x)=sin x,g(x)=cos⁡x and h(x)=cos(cos x), then the integral I =∫ f(g(x)) ⋅ f(x) ⋅ h(x) d x simplifies to -λ (sin)2 (cos ⁡ x)+C (where, C is the constant of integration). The value of λ is equal to (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Given integral, Î= displaystyle ∫ s i n (cos x) . sin â¡ x . c o s (cos â¡ x) d x Let, sin(cos x)=t⇒ -cos(cos â¡ x).sin â¡ xdx=dt ∴ Î= displaystyle ∫ - t d t=-(t2/2)+C =C-(s i n2 (cos x)/2)

Q. If $f (x) = s in x, g (x) = cos x$ and $h (x) = cos (cos x),$ then the integral $I = \int f (g (x)) \cdot f (x) \cdot h (x) d x$ simplifies to $- λ (s in)^{2} (cos x) + C$ (where, $C$ is the constant of integration). The value of $λ$ is equal to

Given integral, $I = \int s in (cos x) . s in x . cos (cos x) d x$
Let, $s in (cos x) = t \Rightarrow - cos (cos x) . s in x d x = d t$
$∴ I = \int - t d t = - \frac{t ^{2}}{2} + C$
$= C - \frac{s i n ^{2} ( cos x )}{2}$

Q. If f(x)=sinx,g(x)=cos⁡x and h(x)=cos(cosx), then the integral I=∫f(g(x))⋅f(x)⋅h(x)dx simplifies to −λ(sin)2(cos⁡x)+C (where, C is the constant of integration). The value of λ is equal to

Given integral, I=∫sin(cosx).sin⁡x.cos(cos⁡x)dx Let, sin(cosx)=t⇒−cos(cos⁡x).sin⁡xdx=dt ∴I=∫−tdt=−2t2​+C =C−2sin2(cosx)​

Q. If $f (x) = s in x, g (x) = cos x$ and $h (x) = cos (cos x),$ then the integral $I = \int f (g (x)) \cdot f (x) \cdot h (x) d x$ simplifies to $- λ (s in)^{2} (cos x) + C$ (where, $C$ is the constant of integration). The value of $λ$ is equal to

Given integral, $I = \int s in (cos x) . s in x . cos (cos x) d x$
Let, $s in (cos x) = t \Rightarrow - cos (cos x) . s in x d x = d t$
$∴ I = \int - t d t = - \frac{t ^{2}}{2} + C$
$= C - \frac{s i n ^{2} ( cos x )}{2}$