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

B. 1/3x^3g(x^3)-∫x^2g(x^3) dx+C. Step 1: Put x^3=t, so 3x^2 dx=dt and x^5 dx=1/3t dt. The integral becomes 1/3∫t f(t) dt. Step 2: By parts with u=t and ∫f(t) dt=g(t): 1/3[t g(t)-∫g(t) dt]. Step 3: Return to x using dt=3x^2 dx: 1/3x^3g(x^3)-∫x^2g(x^3) dx+C. Check: Differentiate option (B): x^2g(x^3)+1/3x^3⋅3x^2f(x^3)-x^2g(x^3)=x^5f(x^3), the integrand. Trap: (C) is not an antiderivative: differentiating it leaves the extra terms x^2g(x^3)-x^3g(x^3), which do not cancel.

Question 3 of 12 · Indefinite Integration

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NIMCET 2023

If $\int f (x) d x = g (x)$ , then $\int x^{5} f (x^{3}) d x$ is equal to

Answer (Detailed Solution Below)View answer ▾

Answer: Option B — $\frac{1}{3} x^{3} g (x^{3}) - \int x^{2} g (x^{3}) d x + C$

Detailed Solution

Put $x^{3} = t$ , so $3 x^{2} d x = d t$ and $x^{5} d x = \frac{1}{3} t d t$ . The integral becomes $\frac{1}{3} \int t f (t) d t$ .

By parts with $u = t$ and $\int f (t) d t = g (t)$ : $\frac{1}{3} [t g (t) - \int g (t) d t]$ .

Return to $x$ using $d t = 3 x^{2} d x$ : $\frac{1}{3} x^{3} g (x^{3}) - \int x^{2} g (x^{3}) d x + C$ .

✓ Check

Differentiate option (B):

x^{2} g (x^{3}) + \frac{1}{3} x^{3} \cdot 3 x^{2} f (x^{3}) - x^{2} g (x^{3}) = x^{5} f (x^{3})

, the integrand.

⚠ Common trap

x^{2} g (x^{3}) - x^{3} g (x^{3})

, which do not cancel.

Quick tip

Put $x^{3} = t$ : $\int x^{5} f (x^{3}) d x = \frac{1}{3} \int t f (t) d t$ , then integrate by parts

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Indefinite Integration — Preparation

This question tests your understanding of Indefinite Integration, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

Indefinite integration is the reverse of differentiation: it finds a function whose derivative is the given function, plus an arbitrary constant C. This topic covers standard integrals, integration by substitution, integration by parts, integration using partial fractions, and integrals of special forms such as 1/(x^2 + a^2) and 1/sqrt(a^2 - x^2). Choosing the right method is the main skill. Look for a function whose derivative also appears in the integrand (substitution), a product of two different types of functions (integration by parts, choosing u with the ILATE rule), or a rational function that can be split (partial fractions). In multiple-choice questions, differentiating the answer options to check which one gives the integrand can save time. Practise the indefinite integration questions below, each with a step-by-step solution.

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If $\int f (x) d x = g (x)$ , then $\int x^{5} f (x^{3}) d x$ is equal to

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