What is the integral of (a+bx^5)/(6ax+bx^6) ?

The integral of (a+bx^5)/(6ax+bx^6) is 1/6 ln|6ax+bx^6|+C

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integral of (a+bx^5)/(6ax+bx^6)

\int \frac{a + b x ^{5}}{6 a x + b x ^{6}} d x

\frac{1}{6} ln 6 a x + b x^{6} + C

Solution steps

\int \frac{a + b x ^{5}}{6 a x + b x ^{6}} d x

Apply u-substitution

: \int \frac{1}{6 u} d u

= \int \frac{1}{6 u} d u

Take the constant out:

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

= \frac{1}{6} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{6} ln ∣ u ∣

Substitute back

u = 6 a x + b x^{6}

= \frac{1}{6} ln 6 a x + b x^{6}

Add a constant to the solution

= \frac{1}{6} ln 6 a x + b x^{6} + C

y^{^{''}} + 4 y^{^{'}} - 24 y = 0

\int e^{x} + e^{x} x d x

g (x) = \frac{1}{x}

\frac{6}{s} + \frac{16}{s ^{2} + 16}

\frac{d ^{2} y}{d x ^{2}} = 0

What is the integral of (a+bx^5)/(6ax+bx^6) ?
The integral of (a+bx^5)/(6ax+bx^6) is 1/6 ln|6ax+bx^6|+C