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인기 있는 미적분학 >

적분 (cos^{23}(x))/(sin(x))

해법

적분

\frac{cos ^{23} ( x )}{sin ( x )}

해법

ln ∣ sin (x) ∣ - \frac{11 sin ^{2} ( x )}{2} + \frac{55 sin ^{4} ( x )}{4} - \frac{55 sin ^{6} ( x )}{2} + \frac{165 sin ^{8} ( x )}{4} - \frac{231 sin ^{10} ( x )}{5} + \frac{77 sin ^{12} ( x )}{2} - \frac{165 sin ^{14} ( x )}{7} + \frac{165 sin ^{16} ( x )}{16} - \frac{55 sin ^{18} ( x )}{18} + \frac{11 sin ^{20} ( x )}{20} - \frac{sin ^{22} ( x )}{22} + C

솔루션 단계

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

삼각성을 사용하여 다시 쓰기

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

대체 적용

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

\frac{( 1 - u ^{2} ) ^{11}}{u}

확대한다:

\frac{1}{u} - 11 u + 55 u^{3} - 165 u^{5} + 330 u^{7} - 462 u^{9} + 462 u^{11} - 330 u^{13} + 165 u^{15} - 55 u^{17} + 11 u^{19} - u^{21}

= \int \frac{1}{u} - 11 u + 55 u^{3} - 165 u^{5} + 330 u^{7} - 462 u^{9} + 462 u^{11} - 330 u^{13} + 165 u^{15} - 55 u^{17} + 11 u^{19} - u^{21} d u

합계 규칙 적용:

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

= \int \frac{1}{u} d u - \int 11 u d u + \int 55 u^{3} d u - \int 165 u^{5} d u + \int 330 u^{7} d u - \int 462 u^{9} d u + \int 462 u^{11} d u - \int 330 u^{13} d u + \int 165 u^{15} d u - \int 55 u^{17} d u + \int 11 u^{19} d u - \int u^{21} d u

\int \frac{1}{u} d u = ln ∣ u ∣

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

\int 55 u^{3} d u = \frac{55 u ^{4}}{4}

\int 165 u^{5} d u = \frac{55 u ^{6}}{2}

\int 330 u^{7} d u = \frac{165 u ^{8}}{4}

\int 462 u^{9} d u = \frac{231 u ^{10}}{5}

\int 462 u^{11} d u = \frac{77 u ^{12}}{2}

\int 330 u^{13} d u = \frac{165 u ^{14}}{7}

\int 165 u^{15} d u = \frac{165 u ^{16}}{16}

\int 55 u^{17} d u = \frac{55 u ^{18}}{18}

\int 11 u^{19} d u = \frac{11 u ^{20}}{20}

\int u^{21} d u = \frac{u ^{22}}{22}

= ln ∣ u ∣ - \frac{11 u ^{2}}{2} + \frac{55 u ^{4}}{4} - \frac{55 u ^{6}}{2} + \frac{165 u ^{8}}{4} - \frac{231 u ^{10}}{5} + \frac{77 u ^{12}}{2} - \frac{165 u ^{14}}{7} + \frac{165 u ^{16}}{16} - \frac{55 u ^{18}}{18} + \frac{11 u ^{20}}{20} - \frac{u ^{22}}{22}

뒤로 대체

u = sin (x)

= ln ∣ sin (x) ∣ - \frac{11 sin ^{2} ( x )}{2} + \frac{55 sin ^{4} ( x )}{4} - \frac{55 sin ^{6} ( x )}{2} + \frac{165 sin ^{8} ( x )}{4} - \frac{231 sin ^{10} ( x )}{5} + \frac{77 sin ^{12} ( x )}{2} - \frac{165 sin ^{14} ( x )}{7} + \frac{165 sin ^{16} ( x )}{16} - \frac{55 sin ^{18} ( x )}{18} + \frac{11 sin ^{20} ( x )}{20} - \frac{sin ^{22} ( x )}{22}

솔루션에 상수 추가

= ln ∣ sin (x) ∣ - \frac{11 sin ^{2} ( x )}{2} + \frac{55 sin ^{4} ( x )}{4} - \frac{55 sin ^{6} ( x )}{2} + \frac{165 sin ^{8} ( x )}{4} - \frac{231 sin ^{10} ( x )}{5} + \frac{77 sin ^{12} ( x )}{2} - \frac{165 sin ^{14} ( x )}{7} + \frac{165 sin ^{16} ( x )}{16} - \frac{55 sin ^{18} ( x )}{18} + \frac{11 sin ^{20} ( x )}{20} - \frac{sin ^{22} ( x )}{22} + C

그래프

인기 있는 예

in t e g r a l \frac{3}{4} - \frac{x}{4}

적분 y= x/(2+1)

in t e g r a l y = \frac{x}{2 + 1}

적분 x/((1+x-x^2))

in t e g r a l \frac{x}{( 1 + x - x ^{2} )}

적분 (x+(x^2+1)^{1/2})^{-1}

in t e g r a l (x + (x^{2} + 1)^{\frac{1}{2}})^{- 1}

in t e g r a l (1 - x)^{5}