TS EAMCET · Maths · Indefinite Integration
\(\int \frac{25 x^2+8}{\sqrt{25 x^2+9}} d x=\)
- A \(\frac{x}{2} \sqrt{25 x^2+9}+\frac{11}{10} \sinh ^{-1}\left(\frac{5 x}{3}\right)+C\)
- B \(\frac{x}{2} \sqrt{25 x^2+9}-\frac{7}{10} \log \left(\frac{5 x+\sqrt{25 x^2+9}}{3}\right)+C\)
- C \(\frac{x}{2} \sqrt{25 x^2+9}+\frac{7}{10} \sinh ^{-1}\left(\frac{5 x}{3}\right)+C\)
- D \(\frac{x}{2} \sqrt{25 x^2+9}+\frac{11}{10} \log \left(\frac{5 x-\sqrt{25 x^2+9}}{3}\right)+C\)
Answer & Solution
Correct Answer
(C) \(\frac{x}{2} \sqrt{25 x^2+9}+\frac{7}{10} \sinh ^{-1}\left(\frac{5 x}{3}\right)+C\)
Step-by-step Solution
Detailed explanation
\(I=\int \frac{25 x^2+8}{\sqrt{25 x^2+9}}=\int\left[\sqrt{25 x^2+9}-\frac{1}{\sqrt{25 x^2+9}}\right] d x\) \(=\frac{x}{2} \sqrt{25 x^2+9}+\frac{9}{2 \times 5} \sinh ^{-1}\left(\frac{5 x}{3}\right)\) \(-\frac{1}{5} \sinh ^{-1}\left(\frac{5 x}{3}\right)+C\)…
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