AP EAMCET · Maths · Indefinite Integration
\(\int \frac{d x}{(1+\sqrt{x})^{2022}}=\)
- A \(\frac{2}{(1+\sqrt{x})^{2021}}\left[\frac{-(1+\sqrt{x})}{2020}+\frac{1}{2021}\right]+C\)
- B \(\frac{2}{(1+\sqrt{x})^{2022}}\left[\frac{1+\sqrt{x}}{2020}-\frac{\sqrt{x}}{2021}\right]+C\)
- C \(\frac{2}{(1+\sqrt{x})}\left[\frac{(1+\sqrt{x})^{2022}}{2022}-\frac{(1+\sqrt{x})^{2021}}{2021}\right]+C\)
- D \(\frac{1}{(1+\sqrt{x})^2}\left[\frac{1}{(1+\sqrt{x})^{1010}}-\frac{1}{(1+\sqrt{x})^{1011}}\right]+C\)
Answer & Solution
Correct Answer
(A) \(\frac{2}{(1+\sqrt{x})^{2021}}\left[\frac{-(1+\sqrt{x})}{2020}+\frac{1}{2021}\right]+C\)
Step-by-step Solution
Detailed explanation
\begin{aligned} I & =\int \frac{d x}{(1+\sqrt{x})^{2022}}, \text { put } x=t^2 \Rightarrow d x=2 t d t \\ \Rightarrow & \int \frac{2 t d t}{(1+t)^{2022}} \\ & =2\left[\int \frac{t+1}{(1+t)^{2022}} d t-\int \frac{1}{(1+t)^{2022}} d t\right] \\ & =2\left[\int…
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