AP EAMCET · Maths · Indefinite Integration
\(\int \frac{e^{2 x}}{\sqrt[4]{e^x+1}} d x=\)
- A \(\frac{4}{7}\left(e^x+1\right)^{4 / 3}\left(3 e^x-1\right)+c\)
- B \(\frac{2}{21}\left(e^x+1\right)^{3 / 4}\left(3 e^x-7\right)+c\)
- C \(\frac{4}{21}\left(e^x+1\right)^{3 / 4}\left(3 e^x-4\right)+c\)
- D \(\frac{8}{21}\left(e^x+1\right)^{3 / 4}\left(3 e^x-1\right)+c\)
Answer & Solution
Correct Answer
(C) \(\frac{4}{21}\left(e^x+1\right)^{3 / 4}\left(3 e^x-4\right)+c\)
Step-by-step Solution
Detailed explanation
\(I=\int \frac{e^{2 x}}{\sqrt[4]{e^x+1}} d x\) Let \(\left(e^x+1\right)^{\frac{1}{4}}=t \Rightarrow e^{2 x}=\left(t^4-1\right)^2\) Differentiating w.r.t. \(x, 2 e^{2 \mathrm{x}} d x=2\left(t^4-1\right) 4 t^3 d t\)…
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