What is the integral of e^{3ln(x)+2x^4} ?

The integral of e^{3ln(x)+2x^4} is 1/8 e^{2x^4}+C

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integral of e^{3ln(x)+2x^4}

\int e^{3 l n (x) + 2 x^{4}} d x

\frac{1}{8} e^{2 x^{4}} + C

Solution steps

\int e^{3 l n (x) + 2 x^{4}} d x

Simplify

e^{3 l n (x) + 2 x^{4}} : e^{2 x^{4}} x^{3}

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

Apply u-substitution

: \int \frac{e ^{u}}{8} d u

= \int \frac{e ^{u}}{8} d u

Take the constant out:

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

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

Use the common integral:

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

= \frac{1}{8} e^{u}

Substitute back

u = 2 x^{4}

= \frac{1}{8} e^{2 x^{4}}

Add a constant to the solution

= \frac{1}{8} e^{2 x^{4}} + C

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What is the integral of e^{3ln(x)+2x^4} ?
The integral of e^{3ln(x)+2x^4} is 1/8 e^{2x^4}+C