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Popular >

integral of sin^{23}(x)

Solution

\int sin^{23} (x) d x

Solution

- cos (x) + \frac{11 cos ^{3} ( x )}{3} - 11 cos^{5} (x) + \frac{165 cos ^{7} ( x )}{7} - \frac{110 cos ^{9} ( x )}{3} + 42 cos^{11} (x) - \frac{462 cos ^{13} ( x )}{13} + 22 cos^{15} (x) - \frac{165 cos ^{17} ( x )}{17} + \frac{55 cos ^{19} ( x )}{19} - \frac{11 cos ^{21} ( x )}{21} + \frac{cos ^{23} ( x )}{23} + C

Solution steps

\int sin^{23} (x) d x

Rewrite using trig identities

= \int (1 - cos^{2} (x))^{11} sin (x) d x

Apply u-substitution

: \int - (1 - u^{2})^{11} d u

= \int - (1 - u^{2})^{11} d u

Expand

- (1 - u^{2})^{11} : - 1 + 11 u^{2} - 55 u^{4} + 165 u^{6} - 330 u^{8} + 462 u^{10} - 462 u^{12} + 330 u^{14} - 165 u^{16} + 55 u^{18} - 11 u^{20} + u^{22}

= \int - 1 + 11 u^{2} - 55 u^{4} + 165 u^{6} - 330 u^{8} + 462 u^{10} - 462 u^{12} + 330 u^{14} - 165 u^{16} + 55 u^{18} - 11 u^{20} + u^{22} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int 1 d u + \int 11 u^{2} d u - \int 55 u^{4} d u + \int 165 u^{6} d u - \int 330 u^{8} d u + \int 462 u^{10} d u - \int 462 u^{12} d u + \int 330 u^{14} d u - \int 165 u^{16} d u + \int 55 u^{18} d u - \int 11 u^{20} d u + \int u^{22} d u

\int 1 d u = u

\int 11 u^{2} d u = \frac{11 u ^{3}}{3}

\int 55 u^{4} d u = 11 u^{5}

\int 165 u^{6} d u = \frac{165 u ^{7}}{7}

\int 330 u^{8} d u = \frac{110 u ^{9}}{3}

\int 462 u^{10} d u = 42 u^{11}

\int 462 u^{12} d u = \frac{462 u ^{13}}{13}

\int 330 u^{14} d u = 22 u^{15}

\int 165 u^{16} d u = \frac{165 u ^{17}}{17}

\int 55 u^{18} d u = \frac{55 u ^{19}}{19}

\int 11 u^{20} d u = \frac{11 u ^{21}}{21}

\int u^{22} d u = \frac{u ^{23}}{23}

= - u + \frac{11 u ^{3}}{3} - 11 u^{5} + \frac{165 u ^{7}}{7} - \frac{110 u ^{9}}{3} + 42 u^{11} - \frac{462 u ^{13}}{13} + 22 u^{15} - \frac{165 u ^{17}}{17} + \frac{55 u ^{19}}{19} - \frac{11 u ^{21}}{21} + \frac{u ^{23}}{23}

Substitute back

u = cos (x)

= - cos (x) + \frac{11 cos ^{3} ( x )}{3} - 11 cos^{5} (x) + \frac{165 cos ^{7} ( x )}{7} - \frac{110 cos ^{9} ( x )}{3} + 42 cos^{11} (x) - \frac{462 cos ^{13} ( x )}{13} + 22 cos^{15} (x) - \frac{165 cos ^{17} ( x )}{17} + \frac{55 cos ^{19} ( x )}{19} - \frac{11 cos ^{21} ( x )}{21} + \frac{cos ^{23} ( x )}{23}

Add a constant to the solution

= - cos (x) + \frac{11 cos ^{3} ( x )}{3} - 11 cos^{5} (x) + \frac{165 cos ^{7} ( x )}{7} - \frac{110 cos ^{9} ( x )}{3} + 42 cos^{11} (x) - \frac{462 cos ^{13} ( x )}{13} + 22 cos^{15} (x) - \frac{165 cos ^{17} ( x )}{17} + \frac{55 cos ^{19} ( x )}{19} - \frac{11 cos ^{21} ( x )}{21} + \frac{cos ^{23} ( x )}{23} + C

Graph

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Frequently Asked Questions (FAQ)

What is the integral of sin^{23}(x) ?
The integral of sin^{23}(x) is