What is the integral of (30x^5)/(2+5x^6) ?

The integral of (30x^5)/(2+5x^6) is ln|2+5x^6|+C

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integral of (30x^5)/(2+5x^6)

\int \frac{30 x ^{5}}{2 + 5 x ^{6}} d x

ln 2 + 5 x^{6} + C

Solution steps

\int \frac{30 x ^{5}}{2 + 5 x ^{6}} d x

Take the constant out:

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

= 30 \cdot \int \frac{x ^{5}}{2 + 5 x ^{6}} d x

Apply u-substitution

: \int \frac{1}{30 u} d u

= 30 \cdot \int \frac{1}{30 u} d u

Take the constant out:

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

= 30 \cdot \frac{1}{30} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 30 \cdot \frac{1}{30} ln ∣ u ∣

Substitute back

u = 2 + 5 x^{6}

= 30 \cdot \frac{1}{30} ln 2 + 5 x^{6}

Simplify

30 \cdot \frac{1}{30} ln 2 + 5 x^{6} : ln 2 + 5 x^{6}

= ln 2 + 5 x^{6}

Add a constant to the solution

= ln 2 + 5 x^{6} + C

\int \frac{x ^{2} - x + 4}{x ^{3} + 2 x} d x

2 sin (3 t) + cos (2 t) + 3 e^{- 2 t} cos (t)

\int \frac{( x - 1 )}{x ^{5}} d x

\frac{\partial}{\partial z} (sin (z))

4 - x^{2}, x^{2} - 4

What is the integral of (30x^5)/(2+5x^6) ?
The integral of (30x^5)/(2+5x^6) is ln|2+5x^6|+C