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

The integral of 4x(x+1)^3 is x(x+1)^4-1/5 (x+1)^5+C

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integral of 4x(x+1)^3

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

x (x + 1)^{4} - \frac{1}{5} (x + 1)^{5} + C

Solution steps

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

Apply Integration By Parts

: \frac{1}{4} x (x + 1)^{4} - \int \frac{1}{4} (x + 1)^{4} d x

= 4 (\frac{1}{4} x (x + 1)^{4} - \int \frac{1}{4} (x + 1)^{4} d x)

\int \frac{1}{4} (x + 1)^{4} d x = \frac{1}{20} (x + 1)^{5}

= 4 (\frac{1}{4} x (x + 1)^{4} - \frac{1}{20} (x + 1)^{5})

Simplify

4 (\frac{1}{4} x (x + 1)^{4} - \frac{1}{20} (x + 1)^{5}) : x (x + 1)^{4} - \frac{1}{5} (x + 1)^{5}

= x (x + 1)^{4} - \frac{1}{5} (x + 1)^{5}

Add a constant to the solution

= x (x + 1)^{4} - \frac{1}{5} (x + 1)^{5} + C

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