What is the integral of (2x)/(sqrt(4x^4-9)) ?

The integral of (2x)/(sqrt(4x^4-9)) is 1/2 ln((|sqrt(4x^4-9)+2x^2|)/3)+C

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integral of (2x)/(sqrt(4x^4-9))

\int \frac{2 x}{4 x ^{4} - 9} d x

\frac{1}{2} ln (\frac{4 x ^{4} - 9 + 2 x ^{2}}{3}) + C

Solution steps

\int \frac{2 x}{4 x ^{4} - 9} d x

Take the constant out:

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

= 2 \cdot \int \frac{x}{4 x ^{4} - 9} d x

Apply u-substitution

= 2 \cdot \int \frac{1}{2 4 u ^{2} - 9} d u

Take the constant out:

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

= 2 \cdot \frac{1}{2} \cdot \int \frac{1}{4 u ^{2} - 9} d u

Apply Trigonometric Substitution

= 2 \cdot \frac{1}{2} \cdot \int \frac{sec ( v )}{2} d v

Take the constant out:

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

= 2 \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \int sec (v) d v

Use the common integral:

\int sec (v) d v = ln ∣ tan (v) + sec (v) ∣

= 2 \cdot \frac{1}{2} \cdot \frac{1}{2} ln ∣ tan (v) + sec (v) ∣

Substitute back

= 2 \cdot \frac{1}{2} \cdot \frac{1}{2} ln tan (arcsec (\frac{2}{3} x^{2})) + sec (arcsec (\frac{2}{3} x^{2}))

Simplify

2 \cdot \frac{1}{2} \cdot \frac{1}{2} ln tan (arcsec (\frac{2}{3} x^{2})) + sec (arcsec (\frac{2}{3} x^{2})) : \frac{1}{2} ln (\frac{4 x ^{4} - 9 + 2 x ^{2}}{3})

= \frac{1}{2} ln (\frac{4 x ^{4} - 9 + 2 x ^{2}}{3})

Add a constant to the solution

= \frac{1}{2} ln (\frac{4 x ^{4} - 9 + 2 x ^{2}}{3}) + C

\frac{d}{d x} (\frac{5}{x + 8})

\int - sin (\frac{1}{2} x) d x

f (x) = 6 cos (x)

\int \frac{tan ^{2} ( x )}{cos ^{2} ( x )} d x

f (x) = 3 x^{3}, (- 1, - 3)

What is the integral of (2x)/(sqrt(4x^4-9)) ?
The integral of (2x)/(sqrt(4x^4-9)) is 1/2 ln((|sqrt(4x^4-9)+2x^2|)/3)+C