Mathematics

# Evaluate $\displaystyle \int \dfrac{1}{\sqrt{25+9x^{2}}}dx$

##### SOLUTION

We have,

$I=\int{\dfrac{1}{\sqrt{25+9{{x}^{2}}}}dx}$

$I=\dfrac{1}{3}\int{\dfrac{1}{\sqrt{\left(\dfrac{5}{3}\right)^2+{{x}^{2}}}}dx}$

So, applying formula,

$\int{\dfrac{1}{\sqrt{{{a}^{2}}+{{x}^{2}}}}dx=\ln \left[ x+\sqrt{{{x}^{2}}+{{a}^{2}}} \right]}$

Therefore,

$I=\dfrac{1}{3}\ln \left[ x+\sqrt{{{x}^{2}}+\left(\dfrac{5}{3}\right)^2} \right]+C$

Hence, this is the answer.

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Subjective Medium Published on 17th 09, 2020
Questions 203525
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