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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