What is the integral of (4x^2+5+2/(x^2+1)) ?

The integral of (4x^2+5+2/(x^2+1)) is (4x^3)/3+5x+2arctan(x)+C

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integral of (4x^2+5+2/(x^2+1))

\int (4 x^{2} + 5 + \frac{2}{x ^{2} + 1}) d x

\frac{4 x ^{3}}{3} + 5 x + 2 arctan (x) + C

Solution steps

\int 4 x^{2} + 5 + \frac{2}{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 x^{2} d x + \int 5 d x + \int \frac{2}{x ^{2} + 1} d x

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

\int 5 d x = 5 x

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

= \frac{4 x ^{3}}{3} + 5 x + 2 arctan (x)

Add a constant to the solution

= \frac{4 x ^{3}}{3} + 5 x + 2 arctan (x) + C

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What is the integral of (4x^2+5+2/(x^2+1)) ?
The integral of (4x^2+5+2/(x^2+1)) is (4x^3)/3+5x+2arctan(x)+C