What is the integral of sqrt(t)+sin(t)-e^{-2t} ?

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

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integral of sqrt(t)+sin(t)-e^{-2t}

\int t + sin (t) - e^{- 2 t} d t

\frac{2}{3} t^{\frac{3}{2}} - cos (t) + \frac{1}{2} e^{- 2 t} + C

Solution steps

\int t + sin (t) - e^{- 2 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 t d t + \int sin (t) d t - \int e^{- 2 t} d t

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

\int sin (t) d t = - cos (t)

\int e^{- 2 t} d t = - \frac{1}{2} e^{- 2 t}

= \frac{2}{3} t^{\frac{3}{2}} - cos (t) - (- \frac{1}{2} e^{- 2 t})

Simplify

= \frac{2}{3} t^{\frac{3}{2}} - cos (t) + \frac{1}{2} e^{- 2 t}

Add a constant to the solution

= \frac{2}{3} t^{\frac{3}{2}} - cos (t) + \frac{1}{2} e^{- 2 t} + C

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