int_0^{x} frac{t^2}{sqrt{a^2+t^2}} dt = | vedclass

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(D) To evaluate $I = \int_0^{x} \frac{t^2}{\sqrt{a^2+t^2}} dt$,we rewrite the numerator as $t^2 = (t^2+a^2) - a^2$. Then,$I = \int_0^{x} \frac{t^2+a^2

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$\frac{x}{2} \sqrt{a^2+x^2} - \frac{a^2}{2} \log \left| \frac{x+\sqrt{a^2+x^2}}{a} \right|$

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(D) To evaluate $I = \int_0^{x} \frac{t^2}{\sqrt{a^2+t^2}} dt$,we rewrite the numerator as $t^2 = (t^2+a^2) - a^2$. Then,$I = \int_0^{x} \frac{t^2+a^2}{\sqrt{a^2+t^2}} dt - \int_0^{x} \frac{a^2}{\sqrt{a^2+t^2}} dt$. $I = \int_0^{x} \sqrt{a^2+t^2} dt - a^2 \int_0^{x} \frac{1}{\sqrt{a^2+t^2}} dt$. Using the standard integrals $\int \sqrt{a^2+t^2} dt = \frac{t}{2}\sqrt{a^2+t^2} + \frac{a^2}{2} \log|t+\sqrt{a^2+t^2}|$ and $\int \frac{1}{\sqrt{a^2+t^2}} dt = \log|t+\sqrt{a^2+t^2}|$: $I = \left[ \frac{t}{2}\sqrt{a^2+t^2} + \frac{a^2}{2} \log|t+\sqrt{a^2+t^2}| - a^2 \log|t+\sqrt{a^2+t^2}| \right]_0^x$. $I = \left[ \frac{t}{2}\sqrt{a^2+t^2} - \frac{a^2}{2} \log|t+\sqrt{a^2+t^2}| \right]_0^x$. Evaluating at limits: $I = \left( \frac{x}{2}\sqrt{a^2+x^2} - \frac{a^2}{2} \log|x+\sqrt{a^2+x^2}| \right) - \left( 0 - \frac{a^2}{2} \log|a| \right)$. $I = \frac{x}{2}\sqrt{a^2+x^2} - \frac{a^2}{2} \log \left| \frac{x+\sqrt{a^2+x^2}}{a} \right|$.

$\int_0^{x} \frac{t^2}{\sqrt{a^2+t^2}} dt =$

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