If f(x),dx = (x), then x^5 f(x^3),dx is equal to

Question

If $\displaystyle\int f(x)\,dx = \psi(x)$, then $\displaystyle\int x^5 f(x^3)\,dx$ is equal to

Accepted Answer

Let $$\int {f\left( x \right)dx = \psi \left( x \right)} $$

Let $$I = \int {{x^5}} f\left( {{x^3}} \right)dx$$

put $${x^3} = t \Rightarrow 3{x^2}dx = dt$$

$$I = {1 \over 3}\int {3.{x^2}} .{x^3}.f\left( {{x^3}} \right).dx$$

$$ = {1 \over 3}\int {tf} \left( t \right)dt$$

$$ = {1 \over 3}\left[ {t\int {f\left( t \right)dt - \int {f\left( t \right)dt} } } \right]$$

$$ = {1 \over 3}\left[ {t\psi \left( t \right) - \int {\psi \left( t \right)dt} } \right]$$

$$ = {1 \over 3}\left[ {{x^3}\psi \left( {{x^3}} \right) - 3\int {{x^2}\psi \left( {{x^3}} \right)dx} } \right] + C$$

$$ = {1 \over 3}{x^3}\psi \left( {{x^3}} \right) - \int {{x^2}} \psi \left( {{x^3}} \right)dx + C$$

Solutions · 1