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

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

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integral of (2t^2+t^2sqrt(t)-1)/(t^2)

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

2 t + \frac{2}{3} t^{\frac{3}{2}} + \frac{1}{t} + C

Solution steps

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

Expand

\frac{2 t ^{2} + t ^{2} t - 1}{t ^{2}} : 2 + t - \frac{1}{t ^{2}}

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

\int 2 d t = 2 t

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

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

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

Simplify

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

Add a constant to the solution

= 2 t + \frac{2}{3} t^{\frac{3}{2}} + \frac{1}{t} + C

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