What is the integral of (7x^2)/(5-x^6) ?

The integral of (7x^2)/(5-x^6) is 7/(6sqrt(5))(ln|(x^3)/(sqrt(5))+1|-ln|(x^3)/(sqrt(5))-1|)+C

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integral of (7x^2)/(5-x^6)

\int \frac{7 x ^{2}}{5 - x ^{6}} d x

\frac{7}{6 5} (ln \frac{x ^{3}}{5} + 1 - ln \frac{x ^{3}}{5} - 1) + C

Solution steps

\int \frac{7 x ^{2}}{5 - x ^{6}} d x

Take the constant out:

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

= 7 \cdot \int \frac{x ^{2}}{5 - x ^{6}} d x

Apply u-substitution

: \int \frac{1}{3 ( 5 - u ^{2} )} d u

= 7 \cdot \int \frac{1}{3 ( 5 - u ^{2} )} d u

Take the constant out:

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

= 7 \cdot \frac{1}{3} \cdot \int \frac{1}{5 - u ^{2}} d u

Apply Integral Substitution

= 7 \cdot \frac{1}{3} \cdot \int \frac{1}{5 ( 1 - v ^{2} )} d v

Take the constant out:

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

= 7 \cdot \frac{1}{3} \cdot \frac{1}{5} \cdot \int \frac{1}{1 - v ^{2}} d v

Use the common integral:

\int \frac{1}{1 - v ^{2}} d v = \frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}

= 7 \cdot \frac{1}{3} \cdot \frac{1}{5} (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2})

Substitute back

= 7 \cdot \frac{1}{3} \cdot \frac{1}{5} \frac{ln \frac{x ^{3}}{5} + 1}{2} - \frac{ln \frac{x ^{3}}{5} - 1}{2}

Simplify

7 \cdot \frac{1}{3} \cdot \frac{1}{5} \frac{ln \frac{x ^{3}}{5} + 1}{2} - \frac{ln \frac{x ^{3}}{5} - 1}{2} : \frac{7}{6 5} (ln \frac{x ^{3}}{5} + 1 - ln \frac{x ^{3}}{5} - 1)

= \frac{7}{6 5} (ln \frac{x ^{3}}{5} + 1 - ln \frac{x ^{3}}{5} - 1)

Add a constant to the solution

= \frac{7}{6 5} (ln \frac{x ^{3}}{5} + 1 - ln \frac{x ^{3}}{5} - 1) + C

x = - 3, x = 1, y = e^{x}, y = 1 - e^{x}

x y^{^{'}} + y = e^{x}, y (1) = 3

x \to 0 lim (x^{2} sin (x^{2}))

\frac{d}{d x} (x^{3} cos (3 x))

\int \frac{8 e ^{8 y}}{6 + e ^{8 y}} d y

What is the integral of (7x^2)/(5-x^6) ?
The integral of (7x^2)/(5-x^6) is 7/(6sqrt(5))(ln|(x^3)/(sqrt(5))+1|-ln|(x^3)/(sqrt(5))-1|)+C