What is the integral of 2x^3e^{x^4+1} ?

The integral of 2x^3e^{x^4+1} is 1/2 e^{x^4+1}+C

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integral of 2x^3e^{x^4+1}

\int 2 x^{3} e^{x^{4} + 1} d x

\frac{1}{2} e^{x^{4} + 1} + C

Solution steps

\int 2 x^{3} e^{x^{4} + 1} d x

Take the constant out:

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

= 2 \cdot \int x^{3} e^{x^{4} + 1} d x

Apply u-substitution

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

= 2 \cdot \int \frac{e ^{u}}{4} d u

Take the constant out:

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

= 2 \cdot \frac{1}{4} \cdot \int e^{u} d u

Use the common integral:

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

= 2 \cdot \frac{1}{4} e^{u}

Substitute back

u = x^{4} + 1

= 2 \cdot \frac{1}{4} e^{x^{4} + 1}

Simplify

2 \cdot \frac{1}{4} e^{x^{4} + 1} : \frac{1}{2} e^{x^{4} + 1}

= \frac{1}{2} e^{x^{4} + 1}

Add a constant to the solution

= \frac{1}{2} e^{x^{4} + 1} + C

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What is the integral of 2x^3e^{x^4+1} ?
The integral of 2x^3e^{x^4+1} is 1/2 e^{x^4+1}+C