What is the integral of t+2+3/(t^2+1) ?

The integral of t+2+3/(t^2+1) is (t^2)/2+2t+3arctan(t)+C

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integral of t+2+3/(t^2+1)

\int t + 2 + \frac{3}{t ^{2} + 1} d t

\frac{t ^{2}}{2} + 2 t + 3 arctan (t) + C

Solution steps

\int t + 2 + \frac{3}{t ^{2} + 1} d t

Apply the Sum Rule:

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

= \int t d t + \int 2 d t + \int \frac{3}{t ^{2} + 1} d t

\int t d t = \frac{t ^{2}}{2}

\int 2 d t = 2 t

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

= \frac{t ^{2}}{2} + 2 t + 3 arctan (t)

Add a constant to the solution

= \frac{t ^{2}}{2} + 2 t + 3 arctan (t) + C

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