What is the integral from 0 to 1 of 2pi(x^5-x^6) ?

The integral from 0 to 1 of 2pi(x^5-x^6) is pi/(21)

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integral from 0 to 1 of 2pi(x^5-x^6)

\int_{0}^{1} 2 π (x^{5} - x^{6}) d x

\frac{π}{21}

Decimal

0.14959 \dots

Solution steps

\int_{0}^{1} 2 π (x^{5} - x^{6}) d x

Take the constant out:

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

= 2 π \cdot \int_{0}^{1} x^{5} - x^{6} 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

= 2 π (\int_{0}^{1} x^{5} d x - \int_{0}^{1} x^{6} d x)

\int_{0}^{1} x^{5} d x = \frac{1}{6}

\int_{0}^{1} x^{6} d x = \frac{1}{7}

= 2 π (\frac{1}{6} - \frac{1}{7})

Simplify

2 π (\frac{1}{6} - \frac{1}{7}) : \frac{π}{21}

= \frac{π}{21}

(x y^{3} + x^{2} y)^{3}

x \to 4 lim (f (x))

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

f (x) = - 4 x

y^{^{''}} - 5 y^{^{'}} + 6 y = cos (t) e^{- t}

What is the integral from 0 to 1 of 2pi(x^5-x^6) ?
The integral from 0 to 1 of 2pi(x^5-x^6) is pi/(21)