What is the integral of (4e^x+1/x+3/(x^2+1)) ?

The integral of (4e^x+1/x+3/(x^2+1)) is 4e^x+ln|x|+3arctan(x)+C

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integral of (4e^x+1/x+3/(x^2+1))

\int (4 e^{x} + \frac{1}{x} + \frac{3}{x ^{2} + 1}) d x

4 e^{x} + ln ∣ x ∣ + 3 arctan (x) + C

Solution steps

\int 4 e^{x} + \frac{1}{x} + \frac{3}{x ^{2} + 1} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 4 e^{x} d x + \int \frac{1}{x} d x + \int \frac{3}{x ^{2} + 1} d x

\int 4 e^{x} d x = 4 e^{x}

\int \frac{1}{x} d x = ln ∣ x ∣

\int \frac{3}{x ^{2} + 1} d x = 3 arctan (x)

= 4 e^{x} + ln ∣ x ∣ + 3 arctan (x)

Add a constant to the solution

= 4 e^{x} + ln ∣ x ∣ + 3 arctan (x) + C

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What is the integral of (4e^x+1/x+3/(x^2+1)) ?
The integral of (4e^x+1/x+3/(x^2+1)) is 4e^x+ln|x|+3arctan(x)+C