QJEEM17Made1e: JEE Main 2017 Mathematics - Class 12 JEEM17Made1e Solution: Let If = , where C is a constant of integration, then the…

Question

Verified Solution • 2025–26 Session moderate

Let

{I_n} = \int {{{\tan }^n}x\,dx} ,\,\left( {n &gt; 1} \right).

If

{I_4} + {I_6}

=

a{\tan ^5}x + b{x^5} + C

, where C is a constant of integration,

then the ordered pair

\left( {a,b} \right)

is equal to

Accepted Answer

Given,

In =

\int {{{\tan }^n}x\,dx,\,\,\,n &gt; 1}

\therefore\,\,\,

I₄ =

\int {{{\tan }^4}x\,dx}

and I₆ =

\int {{{\tan }^6}} x\,dx

\therefore\,\,\,

I = I₄ + I₆

=

\int {\left( {{{\tan }^4}x + {{\tan }^6}x} \right)} dx

=

\int {{{\tan }^4}} x\left( {1 + {{\tan }^2}x} \right)dx

=

\int {{{\tan }^4}} x.{\sec ^2}x\,dx

Let, tanx = t

\Rightarrow

\,\,\,

sec²x dx = dt

\therefore\,\,\,

I =

\int {{t^4}\,dt}

=

{1 \over 5}

t⁵ + C

=

{1 \over 5}

tan⁵x + C

\therefore\,\,\,

By comparing with the question, we get

A =

{1 \over 5}

, B = 0

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