What is the integral of (4x+3)(2x^2+3)^3 ?

The integral of (4x+3)(2x^2+3)^3 is 4x^8+24x^6+54x^4+54x^2+(24x^7)/7+(108x^5)/5+54x^3+81x+C

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integral of (4x+3)(2x^2+3)^3

\int (4 x + 3) (2 x^{2} + 3)^{3} d x

4 x^{8} + 24 x^{6} + 54 x^{4} + 54 x^{2} + \frac{24 x ^{7}}{7} + \frac{108 x ^{5}}{5} + 54 x^{3} + 81 x + C

Solution steps

\int (4 x + 3) (2 x^{2} + 3)^{3} d x

Expand

(4 x + 3) (2 x^{2} + 3)^{3} : 32 x^{7} + 144 x^{5} + 216 x^{3} + 108 x + 24 x^{6} + 108 x^{4} + 162 x^{2} + 81

= \int 32 x^{7} + 144 x^{5} + 216 x^{3} + 108 x + 24 x^{6} + 108 x^{4} + 162 x^{2} + 81 d x

Apply the Sum Rule:

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

= \int 32 x^{7} d x + \int 144 x^{5} d x + \int 216 x^{3} d x + \int 108 x d x + \int 24 x^{6} d x + \int 108 x^{4} d x + \int 162 x^{2} d x + \int 81 d x

\int 32 x^{7} d x = 4 x^{8}

\int 144 x^{5} d x = 24 x^{6}

\int 216 x^{3} d x = 54 x^{4}

\int 108 x d x = 54 x^{2}

\int 24 x^{6} d x = \frac{24 x ^{7}}{7}

\int 108 x^{4} d x = \frac{108 x ^{5}}{5}

\int 162 x^{2} d x = 54 x^{3}

\int 81 d x = 81 x

= 4 x^{8} + 24 x^{6} + 54 x^{4} + 54 x^{2} + \frac{24 x ^{7}}{7} + \frac{108 x ^{5}}{5} + 54 x^{3} + 81 x

Add a constant to the solution

= 4 x^{8} + 24 x^{6} + 54 x^{4} + 54 x^{2} + \frac{24 x ^{7}}{7} + \frac{108 x ^{5}}{5} + 54 x^{3} + 81 x + C

\frac{\partial}{\partial z} (x^{2} y^{3} z)

\frac{d}{d x} (\frac{1}{2 + x})

\frac{d y}{d x} = 2 - \frac{2 x}{1000 - x}

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

y = ln (1 + e^{x})

What is the integral of (4x+3)(2x^2+3)^3 ?
The integral of (4x+3)(2x^2+3)^3 is 4x^8+24x^6+54x^4+54x^2+(24x^7)/7+(108x^5)/5+54x^3+81x+C