AP EAMCET · Maths · Definite Integration
\(\int_0^{\mathrm{x}} \frac{\mathrm{t}^2}{\sqrt{\mathrm{a}^2+\mathrm{t}^2}} \mathrm{dt}=\)
- A \(\frac{x}{2} \sqrt{a^2+x^2}+\log \left|x+\sqrt{a^2+x^2}\right|\)
- B \(\sqrt{a^2+x^2}-a^2 \operatorname{Sinh}^{-1} \frac{x}{a}\)
- C \(\frac{x}{2} \sqrt{a^2+x^2}+\frac{a^2}{4} \log \left|x+\sqrt{a^2+x^2}\right|\)
- D \(\frac{x}{2} \sqrt{a^2+x^2}-\frac{a^2}{2} \operatorname{Sinh}^{-1} \frac{x}{a}\)
Answer & Solution
Correct Answer
(D) \(\frac{x}{2} \sqrt{a^2+x^2}-\frac{a^2}{2} \operatorname{Sinh}^{-1} \frac{x}{a}\)
Step-by-step Solution
Detailed explanation
\( \int_0^{\mathrm{x}} \frac{\mathrm{t}^2}{\sqrt{\mathrm{a}^2+\mathrm{t}^2}} \mathrm{dt} = \int_0^{\mathrm{x}} \frac{\mathrm{t}^2+\mathrm{a}^2-\mathrm{a}^2}{\sqrt{\mathrm{a}^2+\mathrm{t}^2}} \mathrm{dt} \)…
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