What is the integral of e^{1/3 x} ?

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

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integral of e^{1/3 x}

\int e^{\frac{1}{3} x} d x

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

Solution steps

\int e^{\frac{1}{3} x} d x

Apply u-substitution

: \int e^{u} \cdot 3 d u

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

Take the constant out:

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

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

Use the common integral:

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

= 3 e^{u}

Substitute back

u = \frac{1}{3} x

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

Add a constant to the solution

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

x \to - 2 + lim (x + 2)

\frac{d}{d x} (\frac{x}{1 - ln ( x - 1 )})

f (x) = 2 + 4 x^{2}, (0, 2)

y = \frac{8}{sin ( x ) + cos ( x )}, (0, 8)

\frac{\partial}{\partial x} (\frac{3 x - y}{x + 2 y})