What is the integral of x^7e^{x^8} ?

The integral of x^7e^{x^8} is 1/8 e^{x^8}+C

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integral of x^7e^{x^8}

\int x^{7} e^{x^{8}} d x

\frac{1}{8} e^{x^{8}} + C

Solution steps

\int x^{7} e^{x^{8}} d x

Apply u-substitution

: \int \frac{e ^{u^{\frac{8}{7}}} u ^{\frac{1}{7}}}{7} d u

= \int \frac{e ^{u^{\frac{8}{7}}} u ^{\frac{1}{7}}}{7} d u

Take the constant out:

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

= \frac{1}{7} \cdot \int e^{u^{\frac{8}{7}}} u^{\frac{1}{7}} d u

Apply u-substitution

: \int \frac{7 e ^{v}}{8} d v

= \frac{1}{7} \cdot \int \frac{7 e ^{v}}{8} d v

Take the constant out:

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

= \frac{1}{7} \cdot \frac{7}{8} \cdot \int e^{v} d v

Use the common integral:

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

= \frac{1}{7} \cdot \frac{7}{8} e^{v}

Substitute back

= \frac{1}{7} \cdot \frac{7}{8} e^{(x^{7})^{\frac{8}{7}}}

Simplify

\frac{1}{7} \cdot \frac{7}{8} e^{(x^{7})^{\frac{8}{7}}} : \frac{1}{8} e^{x^{8}}

= \frac{1}{8} e^{x^{8}}

Add a constant to the solution

= \frac{1}{8} e^{x^{8}} + C

\int_{1}^{3} \frac{x ^{2} + 2}{4 x - x ^{2}} d x

\int \frac{3 x ^{2}}{x ^{3} + 1} d x

y (x - b) + y^{^{'}} b x = 0

y^{^{'}} = \frac{- y ^{3} + 14 x - 8}{3 x y ^{2}}

\frac{d}{d x} (\frac{2}{2 x ^{4} - 5})

What is the integral of x^7e^{x^8} ?
The integral of x^7e^{x^8} is 1/8 e^{x^8}+C