Question

If $\int f\left(x\right)dx=\Psi \left(x\right)$, then $\int x^{5}f\left(x^3\right)dx$ is equal to

• A. $\frac{1}{3}{x}^3\Psi \left(x^3\right)-\int x^2\Psi \left(x^3\right)dx+C$
• B. $\frac{1}{3}\left[x^3\Psi \right(x^3)-\int x^2\Psi (x^3\left)dx\right]+C$
• C. $\frac{1}{3}{x}^3\Psi \left(x^3\right)-\int x^3\Psi \left(x^3\right)dx+C$
• D. $\frac{1}{3}{x}^3\Psi \left(x^3\right)-3\int x^3\Psi \left(x^3\right)dx+C$

Answer 1

The correct option is A $\frac{1}{3}{x}^3\Psi \left(x^3\right)-\int x^2\Psi \left(x^3\right)dx+C$

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

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

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

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