What is the integral of 5/((x-2)(x^2+4)) ?

The integral of 5/((x-2)(x^2+4)) is 5/4 (1/2 ln| x/2-1|+1/2 (-1/2 ln|(x^2)/4+1|-arctan(x/2)))+C

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integral of 5/((x-2)(x^2+4))

\int \frac{5}{( x - 2 ) ( x ^{2} + 4 )} d x

\frac{5}{4} (\frac{1}{2} ln \frac{x}{2} - 1 + \frac{1}{2} (- \frac{1}{2} ln \frac{x ^{2}}{4} + 1 - arctan (\frac{x}{2}))) + C

Solution steps

\int \frac{5}{( x - 2 ) ( x ^{2} + 4 )} d x

Take the constant out:

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

= 5 \cdot \int \frac{1}{( x - 2 ) ( x ^{2} + 4 )} d x

Apply Integral Substitution

= 5 \cdot \int \frac{1}{4 ( u - 1 ) ( u ^{2} + 1 )} d u

Take the constant out:

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

= 5 \cdot \frac{1}{4} \cdot \int \frac{1}{( u - 1 ) ( u ^{2} + 1 )} d u

Take the partial fraction of

\frac{1}{( u - 1 ) ( u ^{2} + 1 )} : \frac{1}{2 ( u - 1 )} + \frac{- u - 1}{2 ( u ^{2} + 1 )}

= 5 \cdot \frac{1}{4} \cdot \int \frac{1}{2 ( u - 1 )} + \frac{- u - 1}{2 ( u ^{2} + 1 )} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 5 \cdot \frac{1}{4} (\int \frac{1}{2 ( u - 1 )} d u + \int \frac{- u - 1}{2 ( u ^{2} + 1 )} d u)

\int \frac{1}{2 ( u - 1 )} d u = \frac{1}{2} ln ∣ u - 1 ∣

\int \frac{- u - 1}{2 ( u ^{2} + 1 )} d u = \frac{1}{2} (- \frac{1}{2} ln u^{2} + 1 - arctan (u))

= 5 \cdot \frac{1}{4} (\frac{1}{2} ln ∣ u - 1 ∣ + \frac{1}{2} (- \frac{1}{2} ln u^{2} + 1 - arctan (u)))

Substitute back

u = \frac{x}{2}

= 5 \cdot \frac{1}{4} (\frac{1}{2} ln \frac{x}{2} - 1 + \frac{1}{2} (- \frac{1}{2} ln (\frac{x}{2})^{2} + 1 - arctan (\frac{x}{2})))

Simplify

5 \cdot \frac{1}{4} (\frac{1}{2} ln \frac{x}{2} - 1 + \frac{1}{2} (- \frac{1}{2} ln (\frac{x}{2})^{2} + 1 - arctan (\frac{x}{2}))) : \frac{5}{4} (\frac{1}{2} ln \frac{x}{2} - 1 + \frac{1}{2} (- \frac{1}{2} ln \frac{x ^{2}}{4} + 1 - arctan (\frac{x}{2})))

= \frac{5}{4} (\frac{1}{2} ln \frac{x}{2} - 1 + \frac{1}{2} (- \frac{1}{2} ln \frac{x ^{2}}{4} + 1 - arctan (\frac{x}{2})))

Add a constant to the solution

= \frac{5}{4} (\frac{1}{2} ln \frac{x}{2} - 1 + \frac{1}{2} (- \frac{1}{2} ln \frac{x ^{2}}{4} + 1 - arctan (\frac{x}{2}))) + C

\int x^{2} I n d x

lo g_{a} (x)

n = 3 \sum \infty \frac{2}{n ^{2} - 2 n}

\frac{\partial}{\partial y} (- x - y)

x \to - \infty lim (e^{3 x})

What is the integral of 5/((x-2)(x^2+4)) ?
The integral of 5/((x-2)(x^2+4)) is 5/4 (1/2 ln| x/2-1|+1/2 (-1/2 ln|(x^2)/4+1|-arctan(x/2)))+C