Question

If \( \int f(x)\,dx = \Psi(x) \), then \( \int x^5 f(x^3)\,dx = \):

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

Hint:

Use substitution when you see composite functions like \( f(x^3) \). Substituting \( u = x^3 \) simplifies the expression.

Answer 1

The Correct Option is A

We are given: \[ \int f(x)\,dx = \Psi(x) \Rightarrow \int x^5 f(x^3)\,dx = ? \] Let \( u = x^3 \Rightarrow du = 3x^2\,dx \Rightarrow dx = \frac{du}{3x^2} \) Then \( x^5 f(x^3)\,dx = x^5 f(u)\cdot \frac{du}{3x^2} = \frac{1}{3} x^3 f(u)\,du \) So the integral becomes: \[ \int x^5 f(x^3)\,dx = \frac{1}{3} \int x^3 f(x^3)\,du \] Now express \( x^3 \) in terms of \( u \Rightarrow x^3 = u \), so: \[ \frac{1}{3} \int u f(u)\, \frac{du}{u'} \Rightarrow \text{use integration by parts or known reduction formula} \] The result is: \[ \frac{1}{3} x^3 \Psi(x^3) - \int x^2 \Psi(x^3)\,dx \]