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인기 있는 >

integral of sin^4(1/2 x)cos^4(1/2 x)

해법

\int sin^{4} (\frac{1}{2} x) cos^{4} (\frac{1}{2} x) d x

해법

2 (- \frac{1}{4} sin^{3} (\frac{x}{2}) cos (\frac{x}{2}) + \frac{3}{16} (x - sin (x)) - \frac{3}{256} (5 (3 (x - sin (x)) - 4 sin^{3} (\frac{x}{2}) cos (\frac{x}{2})) - 16 sin^{5} (\frac{x}{2}) cos (\frac{x}{2})) - \frac{1}{8} sin^{7} (\frac{x}{2}) cos (\frac{x}{2})) + C

솔루션 단계

\int sin^{4} (\frac{1}{2} x) cos^{4} (\frac{1}{2} x) d x

sin^{4} (\frac{1}{2} x) cos^{4} (\frac{1}{2} x)

단순화하세요:

sin^{4} (\frac{x}{2}) cos^{4} (\frac{x}{2})

= \int sin^{4} (\frac{x}{2}) cos^{4} (\frac{x}{2}) d x

대체 적용

: \int sin^{4} (u) cos^{4} (u) \cdot 2 d u

= \int sin^{4} (u) cos^{4} (u) \cdot 2 d u

정수를 빼라:

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

= 2 \cdot \int sin^{4} (u) cos^{4} (u) d u

삼각성을 사용하여 다시 쓰기

= 2 \cdot \int sin^{4} (u) (1 - sin^{2} (u))^{2} d u

sin^{4} (u) (1 - sin^{2} (u))^{2}

확대한다:

sin^{4} (u) - 2 sin^{6} (u) + sin^{8} (u)

= 2 \cdot \int sin^{4} (u) - 2 sin^{6} (u) + sin^{8} (u) d u

합계 규칙 적용:

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

= 2 (\int sin^{4} (u) d u - \int 2 sin^{6} (u) d u + \int sin^{8} (u) d u)

\int sin^{4} (u) d u = - \frac{1}{4} sin^{3} (u) cos (u) + \frac{3}{8} (u - \frac{1}{2} sin (2 u))

\int 2 sin^{6} (u) d u = 2 (- \frac{cos ( u ) sin ^{5} ( u )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (u) cos (u) + \frac{3}{8} (u - \frac{1}{2} sin (2 u))))

\int sin^{8} (u) d u = - \frac{cos ( u ) sin ^{7} ( u )}{8} + \frac{7}{8} (- \frac{cos ( u ) sin ^{5} ( u )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (u) cos (u) + \frac{3}{8} (u - \frac{1}{2} sin (2 u))))

= 2 (- \frac{1}{4} sin^{3} (u) cos (u) + \frac{3}{8} (u - \frac{1}{2} sin (2 u)) - 2 (- \frac{cos ( u ) sin ^{5} ( u )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (u) cos (u) + \frac{3}{8} (u - \frac{1}{2} sin (2 u)))) - \frac{cos ( u ) sin ^{7} ( u )}{8} + \frac{7}{8} (- \frac{cos ( u ) sin ^{5} ( u )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (u) cos (u) + \frac{3}{8} (u - \frac{1}{2} sin (2 u)))))

뒤로 대체

u = \frac{x}{2}

= 2 (- \frac{1}{4} sin^{3} (\frac{x}{2}) cos (\frac{x}{2}) + \frac{3}{8} (\frac{x}{2} - \frac{1}{2} sin (2 \cdot \frac{x}{2})) - 2 (- \frac{cos ( \frac{x}{2} ) sin ^{5} ( \frac{x}{2} )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (\frac{x}{2}) cos (\frac{x}{2}) + \frac{3}{8} (\frac{x}{2} - \frac{1}{2} sin (2 \cdot \frac{x}{2})))) - \frac{cos ( \frac{x}{2} ) sin ^{7} ( \frac{x}{2} )}{8} + \frac{7}{8} (- \frac{cos ( \frac{x}{2} ) sin ^{5} ( \frac{x}{2} )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (\frac{x}{2}) cos (\frac{x}{2}) + \frac{3}{8} (\frac{x}{2} - \frac{1}{2} sin (2 \cdot \frac{x}{2})))))

2 (- \frac{1}{4} sin^{3} (\frac{x}{2}) cos (\frac{x}{2}) + \frac{3}{8} (\frac{x}{2} - \frac{1}{2} sin (2 \cdot \frac{x}{2})) - 2 (- \frac{cos ( \frac{x}{2} ) sin ^{5} ( \frac{x}{2} )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (\frac{x}{2}) cos (\frac{x}{2}) + \frac{3}{8} (\frac{x}{2} - \frac{1}{2} sin (2 \cdot \frac{x}{2})))) - \frac{cos ( \frac{x}{2} ) sin ^{7} ( \frac{x}{2} )}{8} + \frac{7}{8} (- \frac{cos ( \frac{x}{2} ) sin ^{5} ( \frac{x}{2} )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (\frac{x}{2}) cos (\frac{x}{2}) + \frac{3}{8} (\frac{x}{2} - \frac{1}{2} sin (2 \cdot \frac{x}{2})))))

간소화하다 :

2 (- \frac{1}{4} sin^{3} (\frac{x}{2}) cos (\frac{x}{2}) + \frac{3}{16} (x - sin (x)) - \frac{3}{256} (5 (3 (x - sin (x)) - 4 sin^{3} (\frac{x}{2}) cos (\frac{x}{2})) - 16 sin^{5} (\frac{x}{2}) cos (\frac{x}{2})) - \frac{1}{8} sin^{7} (\frac{x}{2}) cos (\frac{x}{2}))

= 2 (- \frac{1}{4} sin^{3} (\frac{x}{2}) cos (\frac{x}{2}) + \frac{3}{16} (x - sin (x)) - \frac{3}{256} (5 (3 (x - sin (x)) - 4 sin^{3} (\frac{x}{2}) cos (\frac{x}{2})) - 16 sin^{5} (\frac{x}{2}) cos (\frac{x}{2})) - \frac{1}{8} sin^{7} (\frac{x}{2}) cos (\frac{x}{2}))

솔루션에 상수 추가

= 2 (- \frac{1}{4} sin^{3} (\frac{x}{2}) cos (\frac{x}{2}) + \frac{3}{16} (x - sin (x)) - \frac{3}{256} (5 (3 (x - sin (x)) - 4 sin^{3} (\frac{x}{2}) cos (\frac{x}{2})) - 16 sin^{5} (\frac{x}{2}) cos (\frac{x}{2})) - \frac{1}{8} sin^{7} (\frac{x}{2}) cos (\frac{x}{2})) + C

인기 있는 예

f (t) = 4 cos (t)

(y^3-x^3)y^'(x)=3x^2y+1,y(-2)=-1

(y^{3} - x^{3}) y^{^{'}} (x) = 3 x^{2} y + 1, y (- 2) = - 1

y^'=-(1+y)/(1+x+y)

y^{^{'}} = - \frac{1 + y}{1 + x + y}

(dy)/(dx)=7y(7-y)

\frac{d y}{d x} = 7 y (7 - y)

derivative (5x^3+9x^2)/x 유도체

\frac{5 x ^{3} + 9 x ^{2}}{x}