Logarithmic, exponential and trigonometric direct integrals

Compute the indefinite integral $$\int \Big(\dfrac{5}{1+x^2}+3 \sin x +7 x^4+ 2 \cdot 5^x\Big) \ dx$$

We will use the following procedure:

  • Separate the terms into several integrals without taking into account the constant term. $$5 \int\dfrac{1}{1+x^2} \ dx+3\int\sin x \ dx+7\int x^4 \ dx+2\int 5^x \ dx$$

  • Compute each of the integrals separately, applying the formula of direct integrals, and obtain the final result by adding the integration constant. $$5\cdot\arctan(x)-3\cdot\cos(x)+\dfrac{7}{5}x^5+\dfrac{2}{\ln(5)}5^x+C$$

$$ \int \Big(\dfrac{5}{1+x^2}+3 \sin x +7 x^4+ 2 \cdot 5^x\Big) \ dx= 5\cdot\arctan(x)-3\cdot\cos(x)+\dfrac{7}{5}x^5+\dfrac{2}{\ln(5)}5^x+C$$

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