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Question

Evaluate the following integral:
$$\displaystyle \int { \cfrac { { x }^{ 3 }-3{ x }^{ 2 }+5x-7+{ x }^{ 2 }{ a }^{ x } }{ 2{ x }^{ 2 } }  } dx$$


Solution

$$\displaystyle\int{\dfrac{{x}^{3}-3{x}^{2}+5x-7+{x}^{2}{a}^{x}}{2{x}^{2}}dx}$$

$$=\dfrac{1}{2}\displaystyle\int{\dfrac{{x}^{3}-3{x}^{2}+5x-7+{x}^{2}{a}^{x}}{{x}^{2}}dx}$$

$$=\dfrac{1}{2}\displaystyle\int{x\,dx}-\dfrac{3}{2}\displaystyle\int{dx}+\dfrac{5}{2}\displaystyle\int{\dfrac{dx}{x}}-\dfrac{7}{2}\displaystyle\int{\dfrac{dx}{{x}^{2}}}+\dfrac{1}{2}\displaystyle\int{{a}^{x}\,dx}$$

$$=\dfrac{{x}^{2}}{4}-\dfrac{3x}{2}+\dfrac{5}{2}\log{\left|x\right|}+\dfrac{7}{2x}+\dfrac{{a}^{x}}{2\log{a}}+c$$ where $$c$$ is the constant of integration.


Mathematics

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