What is the integral of 4x(4x+1)^3 ?

The integral of 4x(4x+1)^3 is 4((64x^5)/5+12x^4+4x^3+(x^2)/2)+C

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integral of 4x(4x+1)^3

\int 4 x (4 x + 1)^{3} d x

4 (\frac{64 x ^{5}}{5} + 12 x^{4} + 4 x^{3} + \frac{x ^{2}}{2}) + C

Solution steps

\int 4 x (4 x + 1)^{3} d x

Take the constant out:

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

= 4 \cdot \int x (4 x + 1)^{3} d x

Expand

x (4 x + 1)^{3} : 64 x^{4} + 48 x^{3} + 12 x^{2} + x

= 4 \cdot \int 64 x^{4} + 48 x^{3} + 12 x^{2} + x 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

= 4 (\int 64 x^{4} d x + \int 48 x^{3} d x + \int 12 x^{2} d x + \int x d x)

\int 64 x^{4} d x = \frac{64 x ^{5}}{5}

\int 48 x^{3} d x = 12 x^{4}

\int 12 x^{2} d x = 4 x^{3}

\int x d x = \frac{x ^{2}}{2}

= 4 (\frac{64 x ^{5}}{5} + 12 x^{4} + 4 x^{3} + \frac{x ^{2}}{2})

Add a constant to the solution

= 4 (\frac{64 x ^{5}}{5} + 12 x^{4} + 4 x^{3} + \frac{x ^{2}}{2}) + C

What is the integral of 4x(4x+1)^3 ?
The integral of 4x(4x+1)^3 is 4((64x^5)/5+12x^4+4x^3+(x^2)/2)+C