QJEEM19Ma3e53: JEE Main 2019 Mathematics - Class 12 JEEM19Ma3e53 Solution: If   x 5 .e 4x 3 dx = e 4x 3 f(x) + C, where C is a…

Question

Verified Solution • 2025–26 Session moderate

If

\int \,

x⁵.e^{$-$ 4x³} dx =

{1 \over {48}}

e^{$-$ 4x³} f(x) + C, where C is a constant of inegration, then f(x) is equal to -

Accepted Answer

\int {{x^5}} .{e^{ - 4{x^3}}}\,dx = {1 \over {48}}{e^{ - 4{x^3}}}f\left( x \right) + c

Put

{x^3} = t

3{x^2}\,dx = dt

\int {{x^3}.{e^{ - 4{x^3}}}.\,{x^2}} dx

{1 \over 3}\int {t.{e^{ - 4t}}dt}

{1 \over 3}\left[ {t.{{{e^{ - 4t}}} \over { - 4}} - \int {{{{e^{ - 4t}}} \over { - 4}}dt} } \right]

- {{{e^{ - 4t}}} \over {48}}\left[ {4t + 1} \right] + c

{{ - {e^{ - 4{x^3}}}} \over {48}}\left[ {4{x^3} + 1} \right] + c

\therefore

f(x) = - 1 - 4{x^3}

QJEEM19Ma3e53: JEE Main 2019 Mathematics - Class 12 JEEM19Ma3e53 Solution: If &nbsp; x 5 .e 4x 3 dx = e 4x 3 f(x) + C, where C is a…