What is the integral of (y^{n-1})/((ay^n+b)^m) ?

The integral of (y^{n-1})/((ay^n+b)^m) is ((ay^n+b)^{-m+1})/(an(-m+1))+C

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integral of (y^{n-1})/((ay^n+b)^m)

\int \frac{y ^{n - 1}}{( a y ^{n} + b ) ^{m}} d y

\frac{( a y ^{n} + b ) ^{- m + 1}}{an ( - m + 1 )} + C

Solution steps

\int \frac{y ^{n - 1}}{( a y ^{n} + b ) ^{m}} d y

Apply u-substitution

: \int \frac{1}{an u ^{m}} d u

= \int \frac{1}{an u ^{m}} d u

Take the constant out:

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

= \frac{1}{an} \cdot \int \frac{1}{u ^{m}} d u

Apply the Power Rule

: \frac{u ^{- m + 1}}{- m + 1}

= \frac{1}{an} \cdot \frac{u ^{- m + 1}}{- m + 1}

Substitute back

u = a y^{n} + b

= \frac{1}{an} \cdot \frac{( a y ^{n} + b ) ^{- m + 1}}{- m + 1}

Simplify

\frac{1}{an} \cdot \frac{( a y ^{n} + b ) ^{- m + 1}}{- m + 1} : \frac{( a y ^{n} + b ) ^{- m + 1}}{an ( - m + 1 )}

= \frac{( a y ^{n} + b ) ^{- m + 1}}{an ( - m + 1 )}

Add a constant to the solution

= \frac{( a y ^{n} + b ) ^{- m + 1}}{an ( - m + 1 )} + C

f (x) = 10 x

(9, - 2), (- 3, 2)

\int (sin (x) + cos (x)) d x

\int_{4}^{7} x^{4} ln (2 x) d x

x \to 8 lim (\frac{x ^{8}}{x - 8})

What is the integral of (y^{n-1})/((ay^n+b)^m) ?
The integral of (y^{n-1})/((ay^n+b)^m) is ((ay^n+b)^{-m+1})/(an(-m+1))+C