What is the integral of sqrt(3x^2+5) ?

The integral of sqrt(3x^2+5) is 1/2 xsqrt(3x^2+5)+5/(2sqrt(3))ln|sqrt(3/5)x+sqrt(1/5 (5+3x^2))|+C

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integral of sqrt(3x^2+5)

\int 3 x^{2} + 5 d x

\frac{1}{2} x 3 x^{2} + 5 + \frac{5}{2 3} ln \frac{3}{5} x + \frac{1}{5} (5 + 3 x^{2}) + C

Solution steps

\int 3 x^{2} + 5 d x

Apply Trigonometric Substitution

: \int \frac{5}{3} sec^{3} (u) d u

= \int \frac{5}{3} sec^{3} (u) d u

Take the constant out:

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

= \frac{5}{3} \cdot \int sec^{3} (u) d u

Apply Integral Reduction

= \frac{5}{3} (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} \cdot \int sec (u) d u)

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

= \frac{5}{3} (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

Substitute back

u = arctan (\frac{3}{5} x)

= \frac{5}{3} \frac{sec ^{2} ( arctan ( \frac{3}{5} x ) ) sin ( arctan ( \frac{3}{5} x ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{3}{5} x)) + sec (arctan (\frac{3}{5} x))

Simplify

\frac{5}{3} \frac{sec ^{2} ( arctan ( \frac{3}{5} x ) ) sin ( arctan ( \frac{3}{5} x ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{3}{5} x)) + sec (arctan (\frac{3}{5} x)) : \frac{1}{2} x 3 x^{2} + 5 + \frac{5}{2 3} ln \frac{3}{5} x + \frac{1}{5} (5 + 3 x^{2})

= \frac{1}{2} x 3 x^{2} + 5 + \frac{5}{2 3} ln \frac{3}{5} x + \frac{1}{5} (5 + 3 x^{2})

Add a constant to the solution

= \frac{1}{2} x 3 x^{2} + 5 + \frac{5}{2 3} ln \frac{3}{5} x + \frac{1}{5} (5 + 3 x^{2}) + C

\frac{\partial}{\partial x} (9 - x^{3} - y^{5})

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

\frac{d}{d x} (\frac{x}{x ^{2} + 49})

\int x^{- \frac{1}{7}} d x

\frac{1}{x + 4}

What is the integral of sqrt(3x^2+5) ?
The integral of sqrt(3x^2+5) is 1/2 xsqrt(3x^2+5)+5/(2sqrt(3))ln|sqrt(3/5)x+sqrt(1/5 (5+3x^2))|+C