What is the integral of (sqrt(x)-x^3*e^x+x^2)/(x^3) ?

The integral of (sqrt(x)-x^3*e^x+x^2)/(x^3) is -2/(3x^{3/2)}-e^x+ln|x|+C

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integral of (sqrt(x)-x^3*e^x+x^2)/(x^3)

\int \frac{x - x ^{3} \cdot e ^{x} + x ^{2}}{x ^{3}} d x

- \frac{2}{3 x ^{\frac{3}{2}}} - e^{x} + ln ∣ x ∣ + C

Solution steps

\int \frac{x - x ^{3} e ^{x} + x ^{2}}{x ^{3}} d x

Expand

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

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

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

\int e^{x} d x = e^{x}

\int \frac{1}{x} d x = ln ∣ x ∣

= - \frac{2}{3 x ^{\frac{3}{2}}} - e^{x} + ln ∣ x ∣

Add a constant to the solution

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

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