What is the integral of x^{12}e^{x^{13}} ?

The integral of x^{12}e^{x^{13}} is 1/13 e^{x^{13}}+C

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integral of x^{12}e^{x^{13}}

\int x^{12} e^{x^{13}} d x

\frac{1}{13} e^{x^{13}} + C

Solution steps

\int x^{12} e^{x^{13}} d x

Apply u-substitution

: \int \frac{e ^{u^{\frac{13}{12}}} u ^{\frac{1}{12}}}{12} d u

= \int \frac{e ^{u^{\frac{13}{12}}} u ^{\frac{1}{12}}}{12} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{12} \cdot \int e^{u^{\frac{13}{12}}} u^{\frac{1}{12}} d u

Apply u-substitution

: \int \frac{12 e ^{v}}{13} d v

= \frac{1}{12} \cdot \int \frac{12 e ^{v}}{13} d v

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{12} \cdot \frac{12}{13} \cdot \int e^{v} d v

Use the common integral:

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

= \frac{1}{12} \cdot \frac{12}{13} e^{v}

Substitute back

= \frac{1}{12} \cdot \frac{12}{13} e^{(x^{12})^{\frac{13}{12}}}

Simplify

\frac{1}{12} \cdot \frac{12}{13} e^{(x^{12})^{\frac{13}{12}}} : \frac{1}{13} e^{x^{13}}

= \frac{1}{13} e^{x^{13}}

Add a constant to the solution

= \frac{1}{13} e^{x^{13}} + C

f (x) = x^{2} - 9, (- 2, - 5)

\int \frac{x ^{3}}{x ^{4} + 5} d x

\frac{d}{d x} (\frac{1}{x ( 8 + ln ( x ) )})

x \to - 3 lim (\frac{x + 2}{x + 3})

x \to - 3 lim (\frac{x - 2}{x + 5})

What is the integral of x^{12}e^{x^{13}} ?
The integral of x^{12}e^{x^{13}} is 1/13 e^{x^{13}}+C