What is the integral of 8x^7e^{2x^8} ?

The integral of 8x^7e^{2x^8} is 1/2 e^{2x^8}+C

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integral of 8x^7e^{2x^8}

\int 8 x^{7} e^{2 x^{8}} d x

\frac{1}{2} e^{2 x^{8}} + C

Solution steps

\int 8 x^{7} e^{2 x^{8}} d x

Take the constant out:

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

= 8 \cdot \int x^{7} e^{2 x^{8}} d x

Apply u-substitution

: \int \frac{e ^{2 u^{\frac{8}{7}}} u ^{\frac{1}{7}}}{7} d u

= 8 \cdot \int \frac{e ^{2 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

= 8 \cdot \frac{1}{7} \cdot \int e^{2 u^{\frac{8}{7}}} u^{\frac{1}{7}} d u

Apply u-substitution

: \int \frac{7 e ^{2 v}}{8} d v

= 8 \cdot \frac{1}{7} \cdot \int \frac{7 e ^{2 v}}{8} d v

Take the constant out:

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

= 8 \cdot \frac{1}{7} \cdot \frac{7}{8} \cdot \int e^{2 v} d v

Apply u-substitution

: \int e^{w} \frac{1}{2} d w

= 8 \cdot \frac{1}{7} \cdot \frac{7}{8} \cdot \int e^{w} \frac{1}{2} d w

Take the constant out:

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

= 8 \cdot \frac{1}{7} \cdot \frac{7}{8} \cdot \frac{1}{2} \cdot \int e^{w} d w

Use the common integral:

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

= 8 \cdot \frac{1}{7} \cdot \frac{7}{8} \cdot \frac{1}{2} e^{w}

Substitute back

= 8 \cdot \frac{1}{7} \cdot \frac{7}{8} \cdot \frac{1}{2} e^{2 (x^{7})^{\frac{8}{7}}}

Simplify

8 \cdot \frac{1}{7} \cdot \frac{7}{8} \cdot \frac{1}{2} e^{2 (x^{7})^{\frac{8}{7}}} : \frac{1}{2} e^{2 x^{8}}

= \frac{1}{2} e^{2 x^{8}}

Add a constant to the solution

= \frac{1}{2} e^{2 x^{8}} + C

arccos (a x)

\frac{\partial}{\partial x} (x^{2} + y^{2} + 10)

\int_{0}^{\infty} 15 e^{- 5 x} d x

f (x) = 6 x - x^{2}, (1, 5)

f (x) = \frac{1}{x + 8}

What is the integral of 8x^7e^{2x^8} ?
The integral of 8x^7e^{2x^8} is 1/2 e^{2x^8}+C