JEE Mains · Maths · STD 12 - 7.1 indefinite integral
If \(\smallint f\left( x \right)\;dx = \varphi \left( x \right)\), then \(\smallint {x^5}\;f\left( {{x^3}} \right)\;dx = \)
- A \(\frac{1}{3}\left[ {{x^3}\varphi \left( {{x^3}} \right) - \smallint {x^2}\varphi \left( {{x^3}} \right)dx} \right] + c\)
- B \(\frac{1}{3}{x^3}\varphi \left( {{x^3}} \right) - 3\smallint {x^3}\varphi \left( {{x^3}} \right)dx + c\)
- C \(\;\frac{1}{3}{x^3}\varphi \left( {{x^3}} \right) - \smallint {x^2}\varphi \left( {{x^3}} \right)dx + c\)
- D \(\;\frac{1}{3}\left[ {{x^3}\varphi \left( {{x^3}} \right) - \smallint {x^3}\varphi \left( {{x^3}} \right)dx} \right] + c\)
Answer & Solution
Correct Answer
(C) \(\;\frac{1}{3}{x^3}\varphi \left( {{x^3}} \right) - \smallint {x^2}\varphi \left( {{x^3}} \right)dx + c\)
Step-by-step Solution
Detailed explanation
\(\int f(x) d x=\psi(x)\) \(I=\int x^{5} f\left(x^{3}\right) d x\) \(\text { put } x^{3}=t \quad\) \( \Rightarrow \quad x^{2} d x=\frac{d t}{3}\) \(=\frac{1}{3} \int \mathrm{tf}(\mathrm{t}) \mathrm{dt}\)…
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