What is the integral of 3x^3e^{5x^4} ?

The integral of 3x^3e^{5x^4} is 3/20 e^{5x^4}+C

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integral of 3x^3e^{5x^4}

\int 3 x^{3} e^{5 x^{4}} d x

\frac{3}{20} e^{5 x^{4}} + C

Solution steps

\int 3 x^{3} e^{5 x^{4}} d x

Take the constant out:

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

= 3 \cdot \int x^{3} e^{5 x^{4}} d x

Apply u-substitution

: \int \frac{e ^{5 u^{\frac{4}{3}}} u ^{\frac{1}{3}}}{3} d u

= 3 \cdot \int \frac{e ^{5 u^{\frac{4}{3}}} u ^{\frac{1}{3}}}{3} d u

Take the constant out:

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

= 3 \cdot \frac{1}{3} \cdot \int e^{5 u^{\frac{4}{3}}} u^{\frac{1}{3}} d u

Apply u-substitution

: \int \frac{3 e ^{5 v}}{4} d v

= 3 \cdot \frac{1}{3} \cdot \int \frac{3 e ^{5 v}}{4} d v

Take the constant out:

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

= 3 \cdot \frac{1}{3} \cdot \frac{3}{4} \cdot \int e^{5 v} d v

Apply u-substitution

: \int e^{w} \frac{1}{5} d w

= 3 \cdot \frac{1}{3} \cdot \frac{3}{4} \cdot \int e^{w} \frac{1}{5} d w

Take the constant out:

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

= 3 \cdot \frac{1}{3} \cdot \frac{3}{4} \cdot \frac{1}{5} \cdot \int e^{w} d w

Use the common integral:

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

= 3 \cdot \frac{1}{3} \cdot \frac{3}{4} \cdot \frac{1}{5} e^{w}

Substitute back

= 3 \cdot \frac{1}{3} \cdot \frac{3}{4} \cdot \frac{1}{5} e^{5 (x^{3})^{\frac{4}{3}}}

Simplify

3 \cdot \frac{1}{3} \cdot \frac{3}{4} \cdot \frac{1}{5} e^{5 (x^{3})^{\frac{4}{3}}} : \frac{3}{20} e^{5 x^{4}}

= \frac{3}{20} e^{5 x^{4}}

Add a constant to the solution

= \frac{3}{20} e^{5 x^{4}} + C

y^{^{''}} + 3 y^{^{'}} - 10 y = 0

\frac{d}{d x} (e^{x^{4}} \cdot 4 x^{3})

x^{^{''}} + k x = 0

y = \frac{4}{x}, y = 4 x, x = 4, x = 0

\frac{d y}{d x} = 10 - \frac{2 y}{100 - x}

What is the integral of 3x^3e^{5x^4} ?
The integral of 3x^3e^{5x^4} is 3/20 e^{5x^4}+C