What is the integral of (x^5)/(sqrt(4+x^2)) ?

The integral of (x^5)/(sqrt(4+x^2)) is 32(1/2 sqrt(4+x^2)-1/12 (4+x^2)^{3/2}+1/160 (4+x^2)^{5/2})+C

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integral of (x^5)/(sqrt(4+x^2))

\int \frac{x ^{5}}{4 + x ^{2}} d x

32 (\frac{1}{2} 4 + x^{2} - \frac{1}{12} (4 + x^{2})^{\frac{3}{2}} + \frac{1}{160} (4 + x^{2})^{\frac{5}{2}}) + C

Solution steps

\int \frac{x ^{5}}{4 + x ^{2}} d x

Apply Trigonometric Substitution

: \int 32 tan^{5} (u) sec (u) d u

= \int 32 tan^{5} (u) sec (u) d u

Take the constant out:

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

= 32 \cdot \int tan^{5} (u) sec (u) d u

Rewrite using trig identities

= 32 \cdot \int (- 1 + sec^{2} (u))^{2} tan (u) sec (u) d u

Apply u-substitution

: \int (- 1 + v^{2})^{2} d v

= 32 \cdot \int (- 1 + v^{2})^{2} d v

Expand

(- 1 + v^{2})^{2} : 1 - 2 v^{2} + v^{4}

= 32 \cdot \int 1 - 2 v^{2} + v^{4} d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 32 (\int 1 d v - \int 2 v^{2} d v + \int v^{4} d v)

\int 1 d v = v

\int 2 v^{2} d v = \frac{2 v ^{3}}{3}

\int v^{4} d v = \frac{v ^{5}}{5}

= 32 (v - \frac{2 v ^{3}}{3} + \frac{v ^{5}}{5})

Substitute back

= 32 (sec (arctan (\frac{1}{2} x)) - \frac{2 sec ^{3} ( arctan ( \frac{1}{2} x ) )}{3} + \frac{sec ^{5} ( arctan ( \frac{1}{2} x ) )}{5})

Simplify

32 (sec (arctan (\frac{1}{2} x)) - \frac{2 sec ^{3} ( arctan ( \frac{1}{2} x ) )}{3} + \frac{sec ^{5} ( arctan ( \frac{1}{2} x ) )}{5}) : 32 (\frac{1}{2} 4 + x^{2} - \frac{1}{12} (4 + x^{2})^{\frac{3}{2}} + \frac{1}{160} (4 + x^{2})^{\frac{5}{2}})

= 32 (\frac{1}{2} 4 + x^{2} - \frac{1}{12} (4 + x^{2})^{\frac{3}{2}} + \frac{1}{160} (4 + x^{2})^{\frac{5}{2}})

Add a constant to the solution

= 32 (\frac{1}{2} 4 + x^{2} - \frac{1}{12} (4 + x^{2})^{\frac{3}{2}} + \frac{1}{160} (4 + x^{2})^{\frac{5}{2}}) + C

\frac{d ^{2}}{d x ^{2}} (sin (2 x))

\frac{d}{d x} (3 x^{2} y^{2} + 1)

\int e^{0.05 x} d x

n = 1 \sum \infty 1 1^{n} x^{n} n!

\frac{1}{1 + x ^{3}}

What is the integral of (x^5)/(sqrt(4+x^2)) ?
The integral of (x^5)/(sqrt(4+x^2)) is 32(1/2 sqrt(4+x^2)-1/12 (4+x^2)^{3/2}+1/160 (4+x^2)^{5/2})+C