What is the integral from-1 to 0 of (x+2)^{13} ?

The integral from-1 to 0 of (x+2)^{13} is 16383/14

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integral from-1 to 0 of (x+2)^{13}

\int_{- 1}^{0} (x + 2)^{13} d x

\frac{16383}{14}

Decimal

1170.21428 \dots

Solution steps

\int_{- 1}^{0} (x + 2)^{13} d x

Apply Integration By Parts

= [(x + 2)^{13} x - \int 13 (x + 2)^{12} x d x]_{- 1}^{0}

\int 13 (x + 2)^{12} x d x = \frac{13 x ^{14}}{14} + 24 x^{13} + 286 x^{12} + 2080 x^{11} + 10296 x^{10} + 36608 x^{9} + 96096 x^{8} + \frac{1317888 x ^{7}}{7} + 274560 x^{6} + 292864 x^{5} + 219648 x^{4} + 106496 x^{3} + 26624 x^{2}

= [(x + 2)^{13} x - (\frac{13 x ^{14}}{14} + 24 x^{13} + 286 x^{12} + 2080 x^{11} + 10296 x^{10} + 36608 x^{9} + 96096 x^{8} + \frac{1317888 x ^{7}}{7} + 274560 x^{6} + 292864 x^{5} + 219648 x^{4} + 106496 x^{3} + 26624 x^{2})]_{- 1}^{0}

Expand

(x + 2)^{13} x - (\frac{13 x ^{14}}{14} + 24 x^{13} + 286 x^{12} + 2080 x^{11} + 10296 x^{10} + 36608 x^{9} + 96096 x^{8} + \frac{1317888 x ^{7}}{7} + 274560 x^{6} + 292864 x^{5} + 219648 x^{4} + 106496 x^{3} + 26624 x^{2}) : (x + 2)^{13} x - \frac{13 x ^{14}}{14} - 24 x^{13} - 286 x^{12} - 2080 x^{11} - 10296 x^{10} - 36608 x^{9} - 96096 x^{8} - \frac{1317888 x ^{7}}{7} - 274560 x^{6} - 292864 x^{5} - 219648 x^{4} - 106496 x^{3} - 26624 x^{2}

= [x (x + 2)^{13} - \frac{13 x ^{14}}{14} - 24 x^{13} - 286 x^{12} - 2080 x^{11} - 10296 x^{10} - 36608 x^{9} - 96096 x^{8} - \frac{1317888 x ^{7}}{7} - 274560 x^{6} - 292864 x^{5} - 219648 x^{4} - 106496 x^{3} - 26624 x^{2}]_{- 1}^{0}

Compute the boundaries:

\frac{16383}{14}

= \frac{16383}{14}

What is the integral from-1 to 0 of (x+2)^{13} ?
The integral from-1 to 0 of (x+2)^{13} is 16383/14