What is the integral of (2x^3+x^4+sqrt(x))/(x^3) ?

The integral of (2x^3+x^4+sqrt(x))/(x^3) is 2x+(x^2)/2-2/(3x^{3/2)}+C

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integral of (2x^3+x^4+sqrt(x))/(x^3)

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

2 x + \frac{x ^{2}}{2} - \frac{2}{3 x ^{\frac{3}{2}}} + C

Solution steps

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

Expand

\frac{2 x ^{3} + x ^{4} + x}{x ^{3}} : 2 + x + \frac{1}{x ^{\frac{5}{2}}}

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

\int 2 d x = 2 x

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

\int \frac{1}{x ^{\frac{5}{2}}} d x = - \frac{2}{3 x ^{\frac{3}{2}}}

= 2 x + \frac{x ^{2}}{2} - \frac{2}{3 x ^{\frac{3}{2}}}

Add a constant to the solution

= 2 x + \frac{x ^{2}}{2} - \frac{2}{3 x ^{\frac{3}{2}}} + C

What is the integral of (2x^3+x^4+sqrt(x))/(x^3) ?
The integral of (2x^3+x^4+sqrt(x))/(x^3) is 2x+(x^2)/2-2/(3x^{3/2)}+C