What is the integral of (e^x)^3 ?

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

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integral of (e^x)^3

\int (e^{x})^{3} d x

\frac{1}{3} e^{3 x} + C

Solution steps

\int (e^{x})^{3} d x

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= \int e^{3 x} d x

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

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

= \frac{1}{3} e^{u}

Substitute back

u = 3 x

= \frac{1}{3} e^{3 x}

Add a constant to the solution

= \frac{1}{3} e^{3 x} + C

x \to π lim (cos (x - π))

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