What is the integral of x^4e^{x^5+2} ?

The integral of x^4e^{x^5+2} is 1/5 e^{2+x^5}+C

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integral of x^4e^{x^5+2}

\int x^{4} e^{x^{5} + 2} d x

\frac{1}{5} e^{2 + x^{5}} + C

Solution steps

\int x^{4} e^{x^{5} + 2} d x

x^{4} e^{x^{5} + 2} = e^{2} x^{4} e^{x^{5}}

= \int e^{2} x^{4} e^{x^{5}} d x

Take the constant out:

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

= e^{2} \cdot \int x^{4} e^{x^{5}} d x

Apply u-substitution

: \int \frac{e ^{u^{\frac{5}{4}}} u ^{\frac{1}{4}}}{4} d u

= e^{2} \cdot \int \frac{e ^{u^{\frac{5}{4}}} u ^{\frac{1}{4}}}{4} d u

Take the constant out:

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

= e^{2} \frac{1}{4} \cdot \int e^{u^{\frac{5}{4}}} u^{\frac{1}{4}} d u

Apply u-substitution

: \int \frac{4 e ^{v}}{5} d v

= e^{2} \frac{1}{4} \cdot \int \frac{4 e ^{v}}{5} d v

Take the constant out:

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

= e^{2} \frac{1}{4} \cdot \frac{4}{5} \cdot \int e^{v} d v

Use the common integral:

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

= e^{2} \frac{1}{4} \cdot \frac{4}{5} e^{v}

Substitute back

= e^{2} \frac{1}{4} \cdot \frac{4}{5} e^{(x^{4})^{\frac{5}{4}}}

Simplify

e^{2} \frac{1}{4} \cdot \frac{4}{5} e^{(x^{4})^{\frac{5}{4}}} : \frac{1}{5} e^{2 + x^{5}}

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

Add a constant to the solution

= \frac{1}{5} e^{2 + x^{5}} + C

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What is the integral of x^4e^{x^5+2} ?
The integral of x^4e^{x^5+2} is 1/5 e^{2+x^5}+C