If f(x) dx = (x) , then x^5 f(x^3) dx is equal to: - Mathematics MCQ JEE Main 2026 | ExamTest.live

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

Solution & Step-by-step Explanation

Let I = \int x^{5} f (x^{3}) d x . Substitute x^{3} = t \Rightarrow 3 x^{2} d x = d t . I = \int x^{3} \cdot f (x^{3}) \cdot x^{2} d x = \int t \cdot f (t) \cdot \frac{1}{3} d t = \frac{1}{3} \int t f (t) d t Use integration by parts: u = t, d v = f (t) d t \Rightarrow d u = d t, v = Ψ (t) . I = \frac{1}{3} [t Ψ (t) - \int Ψ (t) d t] Substitute t = x^{3} : I = \frac{1}{3} [x^{3} Ψ (x^{3}) - \int Ψ (x^{3}) \cdot 3 x^{2} d x] = \frac{1}{3} x^{3} Ψ (x^{3}) - \int x^{2} Ψ (x^{3}) d x + C Comparing with options, Option C is equivalent.

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

\frac{1}{3} [x^{3} Ψ (x^{3}) - \int x^{3} Ψ (x^{3}) d x] + C

\frac{1}{3} [x^{3} Ψ (x^{3}) - \int 3 x^{2} Ψ (x^{3}) d x] + C

\frac{1}{3} x^{3} Ψ (x^{3}) - \int x^{2} Ψ (x^{3}) d x + C

\frac{1}{3} [x^{3} Ψ (x^{3}) - \int x^{2} Ψ (x^{3}) d x] + C

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