What is the integral of a/(b^{3θ)} ?

The integral of a/(b^{3θ)} is -(ab^{-3θ})/(3ln(b))+C

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integral of a/(b^{3θ)}

\int \frac{a}{b ^{3 θ}} d θ

- \frac{a b ^{- 3 θ}}{3 ln ( b )} + C

Solution steps

\int \frac{a}{b ^{3 θ}} d θ

Take the constant out:

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

= a \cdot \int \frac{1}{b ^{3 θ}} d θ

Apply exponent rule:

\frac{1}{a ^{b}} = a^{- b}

\frac{1}{b ^{3 θ}} = b^{- 3 θ}

= a \cdot \int b^{- 3 θ} d θ

Apply u-substitution

: \int - \frac{1}{3} b^{u} d u

= a \cdot \int - \frac{1}{3} b^{u} d u

Take the constant out:

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

= a (- \frac{1}{3} \cdot \int b^{u} d u)

\int a^{x} d x = \frac{a ^{x}}{ln a}

= a (- \frac{1}{3} \cdot \frac{b ^{u}}{ln ( b )})

Substitute back

u = - 3 θ

= a (- \frac{1}{3} \cdot \frac{b ^{- 3 θ}}{ln ( b )})

Simplify

a (- \frac{1}{3} \cdot \frac{b ^{- 3 θ}}{ln ( b )}) : - \frac{a b ^{- 3 θ}}{3 ln ( b )}

= - \frac{a b ^{- 3 θ}}{3 ln ( b )}

Add a constant to the solution

= - \frac{a b ^{- 3 θ}}{3 ln ( b )} + C

x \to 0 lim (\frac{sin ( x ^{0} )}{x})

\int \frac{x ^{2} - 64}{x ^{2}} d x

x \to 3 lim (\frac{( x ^{3} - 5 x + 6 ) ^{40}}{( x ^{4} - 10 x ^{2} + 9 ) ^{20}})

y = t (ln (10 t))^{2}

x \to 0 lim (3 x - a)

What is the integral of a/(b^{3θ)} ?
The integral of a/(b^{3θ)} is -(ab^{-3θ})/(3ln(b))+C