JEE Mains · Maths · STD 12 - 7.1 indefinite integral
If \(f(x)=\int \frac{1}{x^{1 / 4}\left(1+x^{1 / 4}\right)} \mathrm{d} x, f(0)=-6\), then \(f(1)\) is equal to :
- A \(4\left(\log _e 2-2\right)\)
- B \(2-\log _{e^2} 2\)
- C \(\log _{\mathrm{e}} 2+2\)
- D \(4\left(\log _e 2+2\right)\)
Answer & Solution
Correct Answer
(A) \(4\left(\log _e 2-2\right)\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & \text { Put } x^{1 / 4}=t \Rightarrow d x=4 t^3 d t \\ & \int \frac{4 t^3 d t}{t(t+1)}=4 \int\left(\frac{t^2-1}{t+1}+\frac{1}{t+1}\right) d t \\ & f(x)=4\left[\frac{x^{1 / 2}}{2}-x^{1 / 4}+\ln \left|x^{1 / 4}+1\right|\right]+C\end{aligned}\)…
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