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

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

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

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

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

Solution steps

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

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

\int 2 d x = 2 x

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

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

Add a constant to the solution

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

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