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

The integral of x^3sqrt(1-x) is

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integral of x^3sqrt(1-x)

\int x^{3} 1 - x d x

\frac{1}{4} (1 - x)^{4} 1 - x - (1 - x)^{3} 1 - x + \frac{3}{2} (1 - x)^{2} 1 - x - 1 - x + x 1 - x + \frac{1}{1260} (180 x^{2} + 18 x - 35 (- x + 1)^{3} + 222) (1 - x)^{\frac{3}{2}} + C

Solution steps

\int x^{3} 1 - x d x

Apply u-substitution

: \int - (- u + 1)^{3} u d u

= \int - (- u + 1)^{3} u d u

Take the constant out:

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

= - \int (- u + 1)^{3} u d u

Apply Integration By Parts

: u (- \frac{u ^{4}}{4} + u^{3} - \frac{3 u ^{2}}{2} + u) - \int \frac{1}{8} u (- u^{3} + 4 u^{2} - 6 u + 4) d u

= - (u (- \frac{u ^{4}}{4} + u^{3} - \frac{3 u ^{2}}{2} + u) - \int \frac{1}{8} u (- u^{3} + 4 u^{2} - 6 u + 4) d u)

\int \frac{1}{8} u (- u^{3} + 4 u^{2} - 6 u + 4) d u = \frac{1}{1260} u^{\frac{3}{2}} (- 35 u^{3} + 180 u^{2} - 378 u + 420)

= - (u (- \frac{u ^{4}}{4} + u^{3} - \frac{3 u ^{2}}{2} + u) - \frac{1}{1260} u^{\frac{3}{2}} (- 35 u^{3} + 180 u^{2} - 378 u + 420))

Substitute back

u = 1 - x

= - (1 - x (- \frac{( 1 - x ) ^{4}}{4} + (1 - x)^{3} - \frac{3 ( 1 - x ) ^{2}}{2} + 1 - x) - \frac{1}{1260} (1 - x)^{\frac{3}{2}} (- 35 (1 - x)^{3} + 180 (1 - x)^{2} - 378 (1 - x) + 420))

Simplify

- (1 - x (- \frac{( 1 - x ) ^{4}}{4} + (1 - x)^{3} - \frac{3 ( 1 - x ) ^{2}}{2} + 1 - x) - \frac{1}{1260} (1 - x)^{\frac{3}{2}} (- 35 (1 - x)^{3} + 180 (1 - x)^{2} - 378 (1 - x) + 420)) : \frac{1}{4} (1 - x)^{4} 1 - x - (1 - x)^{3} 1 - x + \frac{3}{2} (1 - x)^{2} 1 - x - 1 - x + x 1 - x + \frac{1}{1260} (180 x^{2} + 18 x - 35 (- x + 1)^{3} + 222) (1 - x)^{\frac{3}{2}}

= \frac{1}{4} (1 - x)^{4} 1 - x - (1 - x)^{3} 1 - x + \frac{3}{2} (1 - x)^{2} 1 - x - 1 - x + x 1 - x + \frac{1}{1260} (180 x^{2} + 18 x - 35 (- x + 1)^{3} + 222) (1 - x)^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{4} (1 - x)^{4} 1 - x - (1 - x)^{3} 1 - x + \frac{3}{2} (1 - x)^{2} 1 - x - 1 - x + x 1 - x + \frac{1}{1260} (180 x^{2} + 18 x - 35 (- x + 1)^{3} + 222) (1 - x)^{\frac{3}{2}} + C

\frac{d}{d x} (4 - \frac{25}{6 x ^{\frac{1}{6}}})

\frac{d}{d x} (ln (3 x^{2} + 7))

f (x) = x - \frac{1}{x}, at x = 1

y^{^{''}} + 81 y = 4 tan (9 x)

x \to 4 lim (\frac{x}{4} + 1)