What is the integral of (7sqrt(x))/(x^3-4) ?

The integral of (7sqrt(x))/(x^3-4) is -7/6 (ln|(x^{3/2})/2+1|-ln|(x^{3/2})/2-1|)+C

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integral of (7sqrt(x))/(x^3-4)

\int \frac{7 x}{x ^{3} - 4} d x

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

Solution steps

\int \frac{7 x}{x ^{3} - 4} 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}{x ^{3} - 4} d x

Apply u-substitution

: \int \frac{2 u ^{2}}{u ^{6} - 4} d u

= 7 \cdot \int \frac{2 u ^{2}}{u ^{6} - 4} d u

Take the constant out:

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

= 7 \cdot 2 \cdot \int \frac{u ^{2}}{u ^{6} - 4} d u

Apply u-substitution

: \int \frac{1}{3 ( v ^{2} - 4 )} d v

= 7 \cdot 2 \cdot \int \frac{1}{3 ( v ^{2} - 4 )} d v

Take the constant out:

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

= 7 \cdot 2 \cdot \frac{1}{3} \cdot \int \frac{1}{v ^{2} - 4} d v

Apply Integral Substitution

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

Take the constant out:

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

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

Factor

w^{2} - 1 : - (- w^{2} + 1)

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

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

Simplify

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

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

Add a constant to the solution

= - \frac{7}{6} (ln \frac{x ^{\frac{3}{2}}}{2} + 1 - ln \frac{x ^{\frac{3}{2}}}{2} - 1) + C

\int (\frac{x - 4}{x ^{9}}) d x

y = (x - 2)^{2}, y = 4 - x, y = 2 x - 2

\frac{d}{d x} (\frac{75}{12 + x ^{2} + y ^{2}})

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

x \to 3 lim (x 18 - x^{2})

What is the integral of (7sqrt(x))/(x^3-4) ?
The integral of (7sqrt(x))/(x^3-4) is -7/6 (ln|(x^{3/2})/2+1|-ln|(x^{3/2})/2-1|)+C