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

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

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integral of e^{x/3}

integral

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

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

Solution steps

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

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

Add a constant to the solution

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

\frac{x ^{2}}{( x + 3 ) ^{3}}

arctan (\frac{1}{2 x})

\frac{( x ^{2} + 4 )}{( x ^{2} - 3 x + 2 )}

\frac{e}{e ^{(x)^{2}}}

\frac{( x - 2 )}{( 2 x + 2 )}