What is the integral of 1/(xsqrt(5x^2-4)) ?

The integral of 1/(xsqrt(5x^2-4)) is 1/2 arcsec((sqrt(5))/2 x)+C

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integral of 1/(xsqrt(5x^2-4))

\int \frac{1}{x 5 x ^{2} - 4} d x

\frac{1}{2} arcsec (\frac{5}{2} x) + C

Solution steps

\int \frac{1}{x 5 x ^{2} - 4} d x

Apply Trigonometric Substitution

: \int \frac{1}{2} d u

= \int \frac{1}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int 1 d u

Integral of a constant:

\int a d x = a x

= \frac{1}{2} \cdot 1 \cdot u

Substitute back

u = arcsec (\frac{5}{2} x)

= \frac{1}{2} \cdot 1 \cdot arcsec (\frac{5}{2} x)

Simplify

= \frac{1}{2} arcsec (\frac{5}{2} x)

Add a constant to the solution

= \frac{1}{2} arcsec (\frac{5}{2} x) + C

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\int \frac{2 t ^{3} - t ^{2} + 2 t + 4}{1 + t ^{2}} d t

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\int (4 x^{3} - 4 x + 6) d x

What is the integral of 1/(xsqrt(5x^2-4)) ?
The integral of 1/(xsqrt(5x^2-4)) is 1/2 arcsec((sqrt(5))/2 x)+C