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Please solve RD Sharma class 12 Chapter Definite integrals exercise 19.2 question 5 maths textbook solution.

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Hint: We use indefinite integral formula then put limits to solve this integral.

Given: \int_{0}^{a}\frac{x}{\sqrt{a^2+x^2}}dx

Solution: \int_{0}^{a}\frac{x}{\sqrt{a^2+x^2}}dx

Putting a^2+x^2=t^2

2x dx=2t dt

x\; \; dx=t\; \; dt

When x=0  then t=a  and when x=a  then t=2a

\begin{aligned} &=\int_{a}^{\sqrt{2} a} \frac{1}{\sqrt{t^{2}}} t d t \\ &=\int_{a}^{\sqrt{2} a} \frac{1}{t} t d t \\ &=\int_{a}^{\sqrt{2} a} 1 d t \\ &=\int_{a}^{\sqrt{2} a} t^{0} d t \\ &=\left[\frac{t^{0+1}}{0+1}\right]_{a}^{\sqrt{2} a} \end{aligned}




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