What is the integral of (3t^2+2e^{3t}+1/t+2cos(3t)) ?

The integral of (3t^2+2e^{3t}+1/t+2cos(3t)) is t^3+2/3 e^{3t}+ln|t|+2/3 sin(3t)+C

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integral of (3t^2+2e^{3t}+1/t+2cos(3t))

\int (3 t^{2} + 2 e^{3 t} + \frac{1}{t} + 2 cos (3 t)) d t

t^{3} + \frac{2}{3} e^{3 t} + ln ∣ t ∣ + \frac{2}{3} sin (3 t) + C

Solution steps

\int 3 t^{2} + 2 e^{3 t} + \frac{1}{t} + 2 cos (3 t) 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 3 t^{2} d t + \int 2 e^{3 t} d t + \int \frac{1}{t} d t + \int 2 cos (3 t) d t

\int 3 t^{2} d t = t^{3}

\int 2 e^{3 t} d t = \frac{2}{3} e^{3 t}

\int \frac{1}{t} d t = ln ∣ t ∣

\int 2 cos (3 t) d t = \frac{2}{3} sin (3 t)

= t^{3} + \frac{2}{3} e^{3 t} + ln ∣ t ∣ + \frac{2}{3} sin (3 t)

Add a constant to the solution

= t^{3} + \frac{2}{3} e^{3 t} + ln ∣ t ∣ + \frac{2}{3} sin (3 t) + C

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What is the integral of (3t^2+2e^{3t}+1/t+2cos(3t)) ?
The integral of (3t^2+2e^{3t}+1/t+2cos(3t)) is t^3+2/3 e^{3t}+ln|t|+2/3 sin(3t)+C