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

The integral of 4x^3e^{4x^4} is 1/4 e^{4x^4}+C

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

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

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

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply u-substitution

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

= 4 \cdot \frac{1}{3} \cdot \frac{3}{4} \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

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

Use the common integral:

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

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

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

y = - \frac{5}{2} x + 3

(3, 2), (2, 3)

x \to 0 lim (\frac{- e ^{\frac{1}{x}}}{x})

\frac{\partial}{\partial x} (3 x ln (x) y)

\int \frac{x - 8}{x ^{2} + 2} d x

What is the integral of 4x^3e^{4x^4} ?
The integral of 4x^3e^{4x^4} is 1/4 e^{4x^4}+C