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

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

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

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

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

Solution steps

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

(3 x + 3) x^{2} + 2 x + 3 = 3 (x + 1) x^{2} + 2 x + 3

= \int 3 (x + 1) x^{2} + 2 x + 3 d x

Take the constant out:

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

= 3 \cdot \int (x + 1) x^{2} + 2 x + 3 d x

Apply u-substitution

: \int u^{2} d u

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

Apply the Power Rule

: \frac{u ^{3}}{3}

= 3 \cdot \frac{u ^{3}}{3}

Substitute back

u = x^{2} + 2 x + 3

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

Simplify

3 \cdot \frac{( x ^{2} + 2 x + 3 ) ^{3}}{3} : (x^{2} + 2 x + 3)^{\frac{3}{2}}

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

Add a constant to the solution

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

n = 1 \sum \infty 9 - \frac{1}{n ^{2}}

n = 1 \sum \infty 10 (- \frac{2}{3})^{n}

n \to \infty lim (\frac{3}{n})

\frac{\partial}{\partial x} (4 x)

ln (sin (2 x))

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