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

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

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

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

\frac{1}{4} e^{4 x^{5}} + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

= 5 \cdot \int \frac{e ^{4 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

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply u-substitution

: \int e^{w} \frac{1}{4} d w

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

Take the constant out:

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

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

Use the common integral:

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

= 5 \cdot \frac{1}{4} \cdot \frac{4}{5} \cdot \frac{1}{4} e^{w}

Substitute back

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

Simplify

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

= \frac{1}{4} e^{4 x^{5}}

Add a constant to the solution

= \frac{1}{4} e^{4 x^{5}} + C

sin (t) - t cos (t)

\frac{d y}{d x}, y = x^{4}

ln (x + 5 + x^{2})

\frac{d}{d x} (\frac{- 4}{x})

x (7 x + 1)^{\frac{1}{3}}

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