What is the integral of x^5e^{x^6+1} ?

The integral of x^5e^{x^6+1} is 1/6 e^{1+x^6}+C

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integral of x^5e^{x^6+1}

\int x^{5} e^{x^{6} + 1} d x

\frac{1}{6} e^{1 + x^{6}} + C

Solution steps

\int x^{5} e^{x^{6} + 1} d x

x^{5} e^{x^{6} + 1} = e^{1} x^{5} e^{x^{6}}

= \int e^{1} x^{5} e^{x^{6}} d x

Take the constant out:

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

= e^{1} \cdot \int x^{5} e^{x^{6}} d x

Simplify

= e \cdot \int x^{5} e^{x^{6}} d x

Apply u-substitution

: \int \frac{e ^{u^{\frac{6}{5}}} u ^{\frac{1}{5}}}{5} d u

= e \cdot \int \frac{e ^{u^{\frac{6}{5}}} u ^{\frac{1}{5}}}{5} d u

Take the constant out:

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

= e \frac{1}{5} \cdot \int e^{u^{\frac{6}{5}}} u^{\frac{1}{5}} d u

Apply u-substitution

: \int \frac{5 e ^{v}}{6} d v

= e \frac{1}{5} \cdot \int \frac{5 e ^{v}}{6} d v

Take the constant out:

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

= e \frac{1}{5} \cdot \frac{5}{6} \cdot \int e^{v} d v

Use the common integral:

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

= e \frac{1}{5} \cdot \frac{5}{6} e^{v}

Substitute back

= e \frac{1}{5} \cdot \frac{5}{6} e^{(x^{5})^{\frac{6}{5}}}

Simplify

e \frac{1}{5} \cdot \frac{5}{6} e^{(x^{5})^{\frac{6}{5}}} : \frac{1}{6} e^{1 + x^{6}}

= \frac{1}{6} e^{1 + x^{6}}

Add a constant to the solution

= \frac{1}{6} e^{1 + x^{6}} + C

∣ x ∣

\frac{d}{d x} (\frac{1}{6} (e^{3 x} + e^{- 3 x}))

\frac{d x}{d t} = 3 x (5 - x)

\frac{\partial}{\partial x} (\frac{x}{x ^{2} - 2 x})

f (x) = \frac{1}{x ^{2}}, (- 1, 1)

What is the integral of x^5e^{x^6+1} ?
The integral of x^5e^{x^6+1} is 1/6 e^{1+x^6}+C