What is the integral of 1/(xsqrt(4x^2-25)) ?

The integral of 1/(xsqrt(4x^2-25)) is 1/5 arcsec(2/5 x)+C

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integral of 1/(xsqrt(4x^2-25))

\int \frac{1}{x 4 x ^{2} - 25} d x

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

Solution steps

\int \frac{1}{x 4 x ^{2} - 25} d x

Apply Trigonometric Substitution

: \int \frac{1}{5} d u

= \int \frac{1}{5} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int 1 d u

Integral of a constant:

\int a d x = a x

= \frac{1}{5} \cdot 1 \cdot u

Substitute back

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

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

Simplify

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

Add a constant to the solution

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

4 \frac{d x}{d t} + 8 x = 16 e^{- 3 t}, x (0) = 10

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

x y^{^{'}} = 4 y

\int \frac{1}{49 e ^{- 4 x} + e ^{4 x}} d x

ln (1 + 4 x), 0

What is the integral of 1/(xsqrt(4x^2-25)) ?
The integral of 1/(xsqrt(4x^2-25)) is 1/5 arcsec(2/5 x)+C