What is the integral of (12x^5)/(3+2x^6) ?

The integral of (12x^5)/(3+2x^6) is ln|3+2x^6|+C

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integral of (12x^5)/(3+2x^6)

\int \frac{12 x ^{5}}{3 + 2 x ^{6}} d x

ln 3 + 2 x^{6} + C

Solution steps

\int \frac{12 x ^{5}}{3 + 2 x ^{6}} d x

Take the constant out:

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

= 12 \cdot \int \frac{x ^{5}}{3 + 2 x ^{6}} d x

Apply u-substitution

: \int \frac{1}{12 u} d u

= 12 \cdot \int \frac{1}{12 u} d u

Take the constant out:

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

= 12 \cdot \frac{1}{12} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 12 \cdot \frac{1}{12} ln ∣ u ∣

Substitute back

u = 3 + 2 x^{6}

= 12 \cdot \frac{1}{12} ln 3 + 2 x^{6}

Simplify

12 \cdot \frac{1}{12} ln 3 + 2 x^{6} : ln 3 + 2 x^{6}

= ln 3 + 2 x^{6}

Add a constant to the solution

= ln 3 + 2 x^{6} + C

\frac{d}{d x} (sin^{4} (2 x))

\frac{d}{d x} (\frac{( x + 1 ) ( 2 x - 7 )}{2 x + 1})

b \to \infty lim (e^{b})

\frac{s \cdot e ^{- 3 s}}{s ^{2} + 25}

y^{^{''}} - 10 y^{^{'}} + 74 y = 0

What is the integral of (12x^5)/(3+2x^6) ?
The integral of (12x^5)/(3+2x^6) is ln|3+2x^6|+C