What is the integral of 10sqrt(5x^2-1) ?

The integral of 10sqrt(5x^2-1) is 5xsqrt(5x^2-1)-sqrt(5)ln|sqrt(5)x+sqrt(5x^2-1)|+C

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integral of 10sqrt(5x^2-1)

\int 10 5 x^{2} - 1 d x

5 x 5 x^{2} - 1 - 5 ln 5 x + 5 x^{2} - 1 + C

Solution steps

\int 10 5 x^{2} - 1 d x

Take the constant out:

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

= 10 \cdot \int 5 x^{2} - 1 d x

Apply Trigonometric Substitution

: \int \frac{1}{5} tan^{2} (u) sec (u) d u

= 10 \cdot \int \frac{1}{5} tan^{2} (u) sec (u) d u

Take the constant out:

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

= 10 \cdot \frac{1}{5} \cdot \int tan^{2} (u) sec (u) d u

Rewrite using trig identities

= 10 \cdot \frac{1}{5} \cdot \int (- 1 + sec^{2} (u)) sec (u) d u

Expand

(- 1 + sec^{2} (u)) sec (u) : - sec (u) + sec^{3} (u)

= 10 \cdot \frac{1}{5} \cdot \int - sec (u) + sec^{3} (u) 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

= 10 \cdot \frac{1}{5} (- \int sec (u) d u + \int sec^{3} (u) d u)

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

\int sec^{3} (u) d u = \frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣

= 10 \cdot \frac{1}{5} (- ln ∣ tan (u) + sec (u) ∣ + \frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

Substitute back

u = arcsec (5 x)

= 10 \cdot \frac{1}{5} (- ln tan (arcsec (5 x)) + sec (arcsec (5 x)) + \frac{1}{2} sec (arcsec (5 x)) tan (arcsec (5 x)) + \frac{1}{2} ln tan (arcsec (5 x)) + sec (arcsec (5 x)))

Simplify

10 \cdot \frac{1}{5} (- ln tan (arcsec (5 x)) + sec (arcsec (5 x)) + \frac{1}{2} sec (arcsec (5 x)) tan (arcsec (5 x)) + \frac{1}{2} ln tan (arcsec (5 x)) + sec (arcsec (5 x))) : 5 x 5 x^{2} - 1 - 5 ln 5 x + 5 x^{2} - 1

= 5 x 5 x^{2} - 1 - 5 ln 5 x + 5 x^{2} - 1

Add a constant to the solution

= 5 x 5 x^{2} - 1 - 5 ln 5 x + 5 x^{2} - 1 + C

\frac{d}{d x} (\frac{2 - \frac{1}{x}}{x - 3})

y^{^{''}} + 6 y^{^{'}} + 9 y = e^{2 x} (3 - 5 x)

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

f (x) = x + 15, at x = 1

(3 x^{2}) d x + (2 y - 4 x^{3} y^{- 1}) d y = 0

What is the integral of 10sqrt(5x^2-1) ?
The integral of 10sqrt(5x^2-1) is 5xsqrt(5x^2-1)-sqrt(5)ln|sqrt(5)x+sqrt(5x^2-1)|+C