Question

If \[ \int f(x) \, dx = g(x), \text{ then } \int x^5 f(x^3) \, dx \] is

• A. \( \frac{1}{3} x^3 g(x^3) - 3 \int x^4 g(x^3) \, dx + c \)
• B. \( \frac{1}{3} x^3 g(x^3) - \int x^2 g(x^3) \, dx + c \)
• C. \( \frac{1}{3} x^3 g(x^3) - \int x^3 g(x^3) \, dx + c \)
• D. None of these

Hint:

When faced with integrals involving a composition of functions, try using substitution and apply the chain rule effectively.

Answer 1

The Correct Option is B

We are given that \( \int f(x) \, dx = g(x) \). This implies that the derivative of \( g(x) \) is \( f(x) \), i.e.,
\[ f(x) = \frac{d}{dx} g(x) \] We are asked to evaluate \( \int x^5 f(x^3) \, dx \). To solve this, let us perform a substitution. Let \( u = x^3 \). Then, \( du = 3x^2 dx \). Rewriting the integral:
\[ \int x^5 f(x^3) \, dx = \int x^2 \cdot x^3 f(x^3) \, dx \] Substituting for \( u \), we get:
\[ \int x^2 \cdot f(x^3) \, dx = \frac{1}{3} \int u^2 f(u) \, du \] Now, using the fact that \( f(u) = \frac{d}{du} g(u) \), we get:
\[ \frac{1}{3} \int u^2 \frac{d}{du} g(u) \, du \] This is equivalent to:
\[ \frac{1}{3} u^3 g(u) - \int u^2 g(u) \, du + c \] Substituting back \( u = x^3 \), we obtain:
\[ \frac{1}{3} x^3 g(x^3) - \int x^2 g(x^3) \, dx + c \] Thus, the correct answer is \( \boxed{(b)} \).