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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int \frac{2 u ^{2}}{3 u ^{2} - 2} d u

= \int \frac{2 u ^{2}}{3 u ^{2} - 2} d u

Take the constant out:

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

= \frac{2}{3} \cdot \int \frac{u ^{2}}{u ^{2} - 2} d u

Apply Trigonometric Substitution

: \int 2 sec^{3} (v) d v

= \frac{2}{3} \cdot \int 2 sec^{3} (v) d v

Take the constant out:

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

= \frac{2}{3} \cdot 2 \cdot \int sec^{3} (v) d v

Apply Integral Reduction

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

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

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

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

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