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

The integral of sqrt(5+3x^2) 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(5+3x^2)

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

\int \frac{5}{9 x ^{2} - 25} d x

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

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

\int 18 x + 37 d x

\int (e^{2 t} + 2)^{\frac{1}{3}} e^{2 t} d t

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