Integrate the following functions-sin x sin cos x d x

Let I =∫ sin x sin (cos x)dx Let cos x =t On differentiating w.r.t.x, we get sin x =dt/dx ⇒ dx =dt/ sin x ∴ I=∫ sin x sin(t)dt/ sin x= ∫ sin t dt = [ cos t]+C ...

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Standard XII

Mathematics

Domain and Range of Basic Inverse Trigonometric Functions

Integrate the...

Question

Integrate the following functions.
$\int$ sin x sin (cos x)dx

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Solution

Let I = $\int$ sin x sin (cos x)dx
Let cos x =t
On differentiating w.r.t.x, we get -sin $x = \frac{d t}{d x}$
$\Rightarrow d x = \frac{d t}{- s i n x}$
$∴ I = \int$ sin x sin(t) $\frac{d t}{- s i n x} = - \int$ sin t dt =-[-cos t]+C =cos (cos x)+C

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Integrate the following functions. ∫ sin x sin (cos x)dx

Let I =∫ sin x sin (cos x)dx Let cos x =t On differentiating w.r.t.x, we get -sin x=dtdx ⇒dx=dt−sinx ∴I=∫ sin x sin(t)dt−sinx=−∫ sin t dt =-[-cos t]+C =cos (cos x)+C

Integrate the following functions.
$\int$ sin x sin (cos x)dx

Let I = $\int$ sin x sin (cos x)dx
Let cos x =t
On differentiating w.r.t.x, we get -sin $x = \frac{d t}{d x}$
$\Rightarrow d x = \frac{d t}{- s i n x}$
$∴ I = \int$ sin x sin(t) $\frac{d t}{- s i n x} = - \int$ sin t dt =-[-cos t]+C =cos (cos x)+C