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

The integral of 4x(x^3-1/4) is 4(1/8 x^2(4x^3-1)-(3x^5)/(10))+C

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integral of 4x(x^3-1/4)

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

4 (\frac{1}{8} x^{2} (4 x^{3} - 1) - \frac{3 x ^{5}}{10}) + C

Solution steps

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

Apply Integration By Parts

: \frac{1}{8} x^{2} (4 x^{3} - 1) - \int \frac{3 x ^{4}}{2} d x

= 4 (\frac{1}{8} x^{2} (4 x^{3} - 1) - \int \frac{3 x ^{4}}{2} d x)

\int \frac{3 x ^{4}}{2} d x = \frac{3 x ^{5}}{10}

= 4 (\frac{1}{8} x^{2} (4 x^{3} - 1) - \frac{3 x ^{5}}{10})

Add a constant to the solution

= 4 (\frac{1}{8} x^{2} (4 x^{3} - 1) - \frac{3 x ^{5}}{10}) + C

f (x) = 1 - 2 x

\frac{x ^{4}}{4}

(y - 2 x - 1) d x + (x + y - 4) d y = 0

n \to \infty lim (4^{n})

x = - 2, x = 3, y = 2 x^{2} + 10, y = 0

What is the integral of 4x(x^3-1/4) ?
The integral of 4x(x^3-1/4) is 4(1/8 x^2(4x^3-1)-(3x^5)/(10))+C