Question

${\text{If }}{{\text{I}}_n} = \smallint {\text{co}}{{\text{s}}^n}xdx{\text{ then }}{{\text{I}}_n} = \\\ {\text{A}}{\text{. }}\dfrac{{ - 1}}{{\text{n}}}{\text{co}}{{\text{s}}^{n - 1}}x{\text{ sinx + }}\left( {\dfrac{{{\text{n}} - 1}}{{\text{n}}}} \right){{\text{I}}_{n - 2}} \\\ {\text{B}}{\text{. co}}{{\text{s}}^{n - 1}}x\sin x + \left( {\dfrac{{{\text{n}} - 1}}{{\text{n}}}} \right){{\text{I}}_{n - 2}} \\\ {\text{C}}{\text{.}}\dfrac{{ - 1}}{{\text{n}}}{\text{co}}{{\text{s}}^{n - 1}}x{\text{ sinx - }}\left( {\dfrac{{{\text{n}} - 1}}{{\text{n}}}} \right){{\text{I}}_{n - 2}} \\\ {\text{D}}{\text{. }}\dfrac{1}{{\text{n}}}{\text{co}}{{\text{s}}^{n - 1}}x{\text{ sinx + }}\left( {\dfrac{{{\text{n}} - 1}}{{\text{n}}}} \right){{\text{I}}_{n - 2}} \\\$

Answer 1

Hint: In this question, first we will break the term ${\text{co}}{{\text{s}}^n}x$ into two parts ‘cosx’ and ‘${\text{co}}{{\text{s}}^{n - 1}}x$. After this, we will apply the ‘By parts’ rule of integration to further solve. We will assume ‘cos x’ as second function and ‘${\text{co}}{{\text{s}}^{n - 1}}x$’ as the first function and then solve it further to get the required expression.

Complete step-by-step answer:
It is given that ${{\text{I}}_n} = \smallint {\text{co}}{{\text{s}}^n}xdx$ . (1)
Let us first divide the function ${\text{co}}{{\text{s}}^n}x$ into two parts ‘cosx’ and ‘${\text{co}}{{\text{s}}^{n - 1}}x$’.
Here we will use the ‘By parts’ rule of integration.
Let ‘cosx’ as second function and ‘${\text{co}}{{\text{s}}^{n - 1}}x$’ as first function.
We know that if ‘u’ and ‘v’ are two functions then using ‘By parts’ rule, we can write:
$\smallint {\text{uvdx = u}}\smallint {\text{vdx - }}\smallint (\dfrac{{{\text{du}}}}{{dx}} \times \smallint {\text{vdx)}}$ , where
‘u’ is taken as first function and
‘v’ is taken as a second function.

Here we are taking ‘cosx’ as second function and ${\text{co}}{{\text{s}}^{n - 1}}x$ as first function
Therefore, equation 1 can be written as:
${{\text{I}}_n} = \smallint \cos x{\text{co}}{{\text{s}}^{n - 1}}xdx = {\text{co}}{{\text{s}}^{n - 1}}x \times \smallint {\text{cosxdx - }}\smallint \left( {\dfrac{{{\text{d(co}}{{\text{s}}^{n - 1}}x)}}{{{\text{dx}}}} \times \smallint {\text{cosxdx}}} \right)$
=${\text{co}}{{\text{s}}^{n - 1}}x \times {\text{sinx - }}\smallint \left( {({\text{n - 1)co}}{{\text{s}}^{n - 2}}x \times - {\text{sinx}} \times {\text{sinxdx}}} \right)$
={\text{co}}{{\text{s}}^{n - 1}}x \times {\text{sinx - }}\left\\{ { - \smallint \left( {({\text{n - 1)co}}{{\text{s}}^{n - 2}}x \times {\text{si}}{{\text{n}}^2}{\text{xdx}}} \right)} \right\\}
=${\text{co}}{{\text{s}}^{n - 1}}x \times {\text{sinx + }}\smallint \left( {({\text{n - 1)co}}{{\text{s}}^{n - 2}}x \times {\text{si}}{{\text{n}}^2}{\text{xdx}}} \right)$
We know that ${\text{si}}{{\text{n}}^2}x = 1 - {\text{co}}{{\text{s}}^2}x$.

    $\therefore $ ${{\text{I}}_n}$ =${\text{co}}{{\text{s}}^{n - 1}}x \times {\text{sinx + }}\smallint \left\\{ {({\text{n - 1)co}}{{\text{s}}^{n - 2}}x(1 - {{\cos }^2}{\text{x)dx}}} \right\\}$  
                   =${\text{co}}{{\text{s}}^{n - 1}}x \times {\text{sinx + (n - 1)}}\smallint \left( {{\text{co}}{{\text{s}}^{n - 2}}x{\text{dx}}} \right) + \left\\{ { - ({\text{n - 1)}}\smallint {\text{co}}{{\text{s}}^{n - 2}}x \times {\text{co}}{{\text{s}}^2}{\text{xdx}}} \right\\}$   
                   =${\text{co}}{{\text{s}}^{n - 1}}x \times {\text{sinx + (n - 1)}}\smallint \left( {{\text{co}}{{\text{s}}^{n - 2}}x{\text{dx}}} \right) - ({\text{n - 1)}}\smallint {\text{co}}{{\text{s}}^n}x{\text{dx}}$.  

But we know that ${{\text{I}}_n} = \smallint {\text{co}}{{\text{s}}^n}xdx$ .
$\therefore {{\text{I}}_n} = {\text{co}}{{\text{s}}^{n - 1}}x \times {\text{sinx + (n - 1)}}\smallint \left( {{\text{co}}{{\text{s}}^{n - 2}}x{\text{dx}}} \right) - ({\text{n - 1)}}{{\text{I}}_n}$
Also, ${{\text{I}}_{n - 2}} = \smallint {\text{co}}{{\text{s}}^{n - 2}}xdx$.
$\therefore {{\text{I}}_n} + ({\text{n - 1)}}{{\text{I}}_n} = {\text{co}}{{\text{s}}^{n - 1}}x \times {\text{sinx + (n - 1)}}{{\text{I}}_{n - 2}}$
\Rightarrow {{\text{I}}_n}\left\\{ {1 + ({\text{n - 1)}}} \right\\} = {\text{co}}{{\text{s}}^{n - 1}}x \times {\text{sinx + (n - 1)}}{{\text{I}}_{n - 2}}
$\Rightarrow {{\text{I}}_n} \times {\text{n}} = {\text{co}}{{\text{s}}^{n - 1}}x \times {\text{sinx + (n - 1)}}{{\text{I}}_{n - 2}}$
$\Rightarrow {{\text{I}}_n} = \dfrac{{{\text{co}}{{\text{s}}^{n - 1}}x \times {\text{sinx + (n - 1)}}{{\text{I}}_{n - 2}}}}{{\text{n}}} = \dfrac{{{\text{co}}{{\text{s}}^{n - 1}}x \times {\text{sinx}}}}{{\text{n}}} + \dfrac{{({\text{n - 1)}}}}{{\text{n}}}{{\text{I}}_{n - 2}}$
So, option D is correct.

Note: In this question, the important step is division of a given function into two parts. You should remember the method of solving integration using the ‘By part’ rule. Here the first and second function are generally chosen according to ILATE rule, where:
I- Inverse function
L- Logarithmic function
A- Algebraic function
T- Trigonometric function
E- Exponential function
Second function is chosen in decreasing order from top to bottom.