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

The integral from 0 to 1 of 2pix(x^5-x^6) is pi/(28)

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

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

\frac{π}{28}

Decimal Notation

0.11219 \dots

Solution steps

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

Expand

x (x^{5} - x^{6}) : x^{6} - x^{7}

= 2 π \cdot \int_{0}^{1} x^{6} - x^{7} 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^{6} d x - \int_{0}^{1} x^{7} d x)

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

\int_{0}^{1} x^{7} d x = \frac{1}{8}

= 2 π (\frac{1}{7} - \frac{1}{8})

Simplify

2 π (\frac{1}{7} - \frac{1}{8}) : \frac{π}{28}

= \frac{π}{28}

\int \frac{1}{e ^{2 y}} d y

x \frac{d y}{d x} + 2 y = x^{3} - x

\int_{0}^{π} t cos^{2} (t) d t

\int sin (\frac{nπ x}{1}) d x

x \to 1 + lim (x + 1)

What is the integral from 0 to 1 of 2pix(x^5-x^6) ?
The integral from 0 to 1 of 2pix(x^5-x^6) is pi/(28)