AP EAMCET · Maths · Definite Integration
If \(\int \frac{x^3}{\sqrt{1+x^2}} d x=A\left(1+x^2\right)^{\frac{3}{2}}+B\left(1+x^2\right)^{\frac{1}{2}}+C\), then \(\mathrm{A}+\mathrm{B}=\)
- A \(\frac{2}{3}\)
- B \(-\frac{2}{3}\)
- C \(\frac{1}{3}\)
- D \(-\frac{1}{3}\)
Answer & Solution
Correct Answer
(B) \(-\frac{2}{3}\)
Step-by-step Solution
Detailed explanation
\(\int \frac{x^3}{\sqrt{1+x^2}} d x=\mathrm{A}\left(1+x^2\right)^{3 / 2}+\mathrm{B}\left(1+x^2\right)^{1 / 2}+c\) ...(i) \(\int \frac{x^3}{\sqrt{1+x^2}} d x=\int \frac{x^2 \cdot x}{\sqrt{1+x^2}} d x\) Let \(1+x^2=t^2 \Rightarrow x d x=t d t\)…
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