What is the integral of sqrt(x(x+1)) ?

The integral of sqrt(x(x+1)) is 1/8 (4xsqrt(x(x+1))+2sqrt(x(x+1))-ln|2x+1+2sqrt(x(x+1))|)+C

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integral of sqrt(x(x+1))

\int x (x + 1) d x

\frac{1}{8} (4 x x (x + 1) + 2 x (x + 1) - ln 2 x + 1 + 2 x (x + 1)) + C

Solution steps

\int x (x + 1) d x

Apply u-substitution

: \int (u - 1) u d u

= \int (u - 1) u d u

Expand

(u - 1) u : u^{2} - u

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

Complete the square

u^{2} - u : (u - \frac{1}{2})^{2} - \frac{1}{4}

= \int (u - \frac{1}{2})^{2} - \frac{1}{4} d u

Apply u-substitution

: \int \frac{4 v ^{2} - 1}{2} d v

= \int \frac{4 v ^{2} - 1}{2} d v

Take the constant out:

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

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

Apply Trigonometric Substitution

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

= \frac{1}{2} \cdot \int \frac{1}{2} tan^{2} (w) sec (w) d w

Take the constant out:

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

= \frac{1}{2} \cdot \frac{1}{2} \cdot \int tan^{2} (w) sec (w) d w

Rewrite using trig identities

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

Expand

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

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

Apply the Sum Rule:

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

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

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

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

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

Substitute back

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

Simplify

\frac{1}{2} \cdot \frac{1}{2} (- ln tan (arcsec (2 (x + 1 - \frac{1}{2}))) + sec (arcsec (2 (x + 1 - \frac{1}{2}))) + \frac{1}{2} sec (arcsec (2 (x + 1 - \frac{1}{2}))) tan (arcsec (2 (x + 1 - \frac{1}{2}))) + \frac{1}{2} ln tan (arcsec (2 (x + 1 - \frac{1}{2}))) + sec (arcsec (2 (x + 1 - \frac{1}{2})))) : \frac{1}{8} (4 x x (x + 1) + 2 x (x + 1) - ln 2 x + 1 + 2 x (x + 1))

= \frac{1}{8} (4 x x (x + 1) + 2 x (x + 1) - ln 2 x + 1 + 2 x (x + 1))

Add a constant to the solution

= \frac{1}{8} (4 x x (x + 1) + 2 x (x + 1) - ln 2 x + 1 + 2 x (x + 1)) + C

\int_{1}^{100} \frac{1}{x ^{2}} d x

\frac{d y}{d x}, 2 x = y^{4}

x \to 0 lim (\frac{x}{- 4 x ^{3} - 8 x})

\frac{\partial}{\partial t} (\frac{( s - 3 r )}{t})

\int 2 x (x^{2} + 1)^{4} d x

What is the integral of sqrt(x(x+1)) ?
The integral of sqrt(x(x+1)) is 1/8 (4xsqrt(x(x+1))+2sqrt(x(x+1))-ln|2x+1+2sqrt(x(x+1))|)+C