$\int\left\{\frac{x}{a}+\frac{b}{x}+x^a+b^x+a b\right\} d x$ is equal to

$\int\left\{\frac{x}{a}+\frac{b}{x}+x^a+b^x+a b\right\} d x$ is equal to
  1. $\frac{x^2}{2 a}+\frac{b}{x^2}+\frac{x^{a+1}}{a+1}+\frac{b^x}{\log b}+C$
  2. $\frac{x^2}{2 a}+b \log |x|+\frac{x^{a+1}}{a+1}+\frac{b^x}{\log b}+a b x+C$
  3. $\frac{1}{a}+b \log |x|+a x^{a-1}+b^x \log b+a b+C$
  4. $\frac{x^2}{2 a}+b \log |x|+\frac{x^{a+1}}{a+1}+\frac{b^x}{\log a}+a b x+C$

Solution

$ \begin{aligned} & \int\left(\frac{x}{a}+b x^{-1}+x^a+b^x+a b\right) d x \\ & =\frac{1}{a} \cdot \frac{x^2}{2}+b \log x+\frac{x^{a+1}}{a+1}+b^x \log b+a b x+C \end{aligned} $

Asked in: AP EAMCET 2021 (23 Aug Shift 1)

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