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  • Provide Solution For R.D.Sharma Maths Class 12 Chapter 18 Indefinite Integrals Exercise Revision Exercise Question 74 Maths Textbook Solution.

Provide Solution For  R.D.Sharma Maths Class 12 Chapter 18  Indefinite Integrals Exercise  Revision Exercise Question 74 Maths Textbook Solution.

Answers (1)

Answer:

 \sqrt{x^{2}+a x}+\frac{a}{2} \log \left|\left(x+\frac{a}{2}\right)+\sqrt{x^{2}+a x}\right|+c

Hint:

To solve the given statement multiply and divide the equation by a +x.

Given:

\int \sqrt{\frac{a+x}{x}} d x

Solution:

=\int \frac{a+x}{\sqrt{x(a+x)}} d x

=\frac{1}{2} \int \frac{2(a+x)}{\sqrt{x^{2}+a x}} d x

=\frac{1}{2}\left[\int \frac{2 x+a}{\sqrt{x^{2}+a x}} d x+\int \frac{a}{\sqrt{x^{2}+a x}} d x\right]

\left[d\left(\sqrt{x^{2}+a x}\right)=\frac{-1}{2 \sqrt{x^{2}+a x}}(2 x+a)\right]

\left[d\left(\sqrt{x^{2}+a x}\right)=\frac{a}{\sqrt{x^{2}+a x+\left(\frac{a^{2}}{4}\right)-\left(\frac{a^{2}}{4}\right)}}\right]

\left[d\left(\sqrt{x^{2}+a x}\right)=\frac{a}{\sqrt{\left(x+\frac{a}{2}\right)^{2}-\left(\frac{a}{2}\right)^{2}}}\right]

=\frac{1}{2} \cdot 2 \sqrt{x^{2}+a x}+\int \frac{a}{\sqrt{\left(x+\frac{a}{2}\right)^{2}-\left(\frac{a}{2}\right)}} d x

=\sqrt{x^{2}+a x}+a \log \left|\left(x+\frac{a}{2}\right)+\sqrt{x^{2}+a x}\right|

=\sqrt{x^{2}+a x}+\frac{a}{2} \log \left|\left(x+\frac{a}{2}\right)+\sqrt{x^{2}+a x}\right|+c

 

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