What is the integral of 1/(4+5x^6)(30x^5) ?

The integral of 1/(4+5x^6)(30x^5) is ln|4+5x^6|+C

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integral of 1/(4+5x^6)(30x^5)

\int \frac{1}{4 + 5 x ^{6}} (30 x^{5}) d x

ln 4 + 5 x^{6} + C

Solution steps

\int \frac{1}{4 + 5 x ^{6}} \cdot 30 x^{5} d x

Take the constant out:

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

= 30 \cdot \int \frac{1}{4 + 5 x ^{6}} x^{5} 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 = 4 + 5 x^{6}

= 30 \cdot \frac{1}{30} ln 4 + 5 x^{6}

Simplify

30 \cdot \frac{1}{30} ln 4 + 5 x^{6} : ln 4 + 5 x^{6}

= ln 4 + 5 x^{6}

Add a constant to the solution

= ln 4 + 5 x^{6} + C

x (2 x + 7)^{\frac{1}{3}}

t^{2} y^{^{''}} + 6 t y^{^{'}} - 6 y = 0

\frac{\partial}{\partial x} ((x^{3} - y^{2})^{(- 1)})

\frac{d}{d x} (4 x^{2} + 3 x + 2)

y = e^{e}

What is the integral of 1/(4+5x^6)(30x^5) ?
The integral of 1/(4+5x^6)(30x^5) is ln|4+5x^6|+C