What is the integral of (x^6e^x-5x^5)/(x^6) ?

The integral of (x^6e^x-5x^5)/(x^6) is e^x-5ln|x|+C

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integral of (x^6e^x-5x^5)/(x^6)

\int \frac{x ^{6} e ^{x} - 5 x ^{5}}{x ^{6}} d x

e^{x} - 5 ln ∣ x ∣ + C

Solution steps

\int \frac{x ^{6} e ^{x} - 5 x ^{5}}{x ^{6}} d x

Expand

\frac{x ^{6} e ^{x} - 5 x ^{5}}{x ^{6}} : e^{x} - \frac{5}{x}

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

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

\int \frac{5}{x} d x = 5 ln ∣ x ∣

= e^{x} - 5 ln ∣ x ∣

Add a constant to the solution

= e^{x} - 5 ln ∣ x ∣ + C

What is the integral of (x^6e^x-5x^5)/(x^6) ?
The integral of (x^6e^x-5x^5)/(x^6) is e^x-5ln|x|+C