What is the integral of 1/(2+5x^6)(30x^5) ?

The integral of 1/(2+5x^6)(30x^5) is ln|2+5x^6|+C

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integral of 1/(2+5x^6)(30x^5)

\int \frac{1}{2 + 5 x ^{6}} (30 x^{5}) d x

ln 2 + 5 x^{6} + C

Solution steps

\int \frac{1}{2 + 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}{2 + 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 = 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{1}{25 - x ^{2}} d x

f (x) = 9 x^{2} + 6 x - 6

7 π x^{2}

x \to 0 + lim (12 e^{- \frac{2}{x}})

f (R) = R^{3}

What is the integral of 1/(2+5x^6)(30x^5) ?
The integral of 1/(2+5x^6)(30x^5) is ln|2+5x^6|+C