integral of sin^4(θ)cos^4(θ)

解答

\int sin^{4} (θ) cos^{4} (θ) d θ

- \frac{1}{4} sin^{3} (θ) cos (θ) + \frac{3}{8} (θ - \frac{1}{2} sin (2 θ)) - \frac{9}{8} (- \frac{1}{6} sin^{5} (θ) cos (θ) + \frac{5}{6} (- \frac{1}{4} sin^{3} (θ) cos (θ) + \frac{3}{8} (θ - \frac{1}{2} sin (2 θ)))) - \frac{1}{8} sin^{7} (θ) cos (θ) + C

求解步骤

\int sin^{4} (θ) cos^{4} (θ) d θ

使用三角恒等式改写

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

乘开

sin^{4} (θ) (1 - sin^{2} (θ))^{2} : sin^{4} (θ) - 2 sin^{6} (θ) + sin^{8} (θ)

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

使用积分加法定则:

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

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

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

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

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

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

化简

- \frac{1}{4} sin^{3} (θ) cos (θ) + \frac{3}{8} (θ - \frac{1}{2} sin (2 θ)) - 2 (- \frac{cos ( θ ) sin ^{5} ( θ )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (θ) cos (θ) + \frac{3}{8} (θ - \frac{1}{2} sin (2 θ)))) - \frac{cos ( θ ) sin ^{7} ( θ )}{8} + \frac{7}{8} (- \frac{cos ( θ ) sin ^{5} ( θ )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (θ) cos (θ) + \frac{3}{8} (θ - \frac{1}{2} sin (2 θ)))) : - \frac{1}{4} sin^{3} (θ) cos (θ) + \frac{3}{8} (θ - \frac{1}{2} sin (2 θ)) - \frac{9}{8} (- \frac{1}{6} sin^{5} (θ) cos (θ) + \frac{5}{6} (- \frac{1}{4} sin^{3} (θ) cos (θ) + \frac{3}{8} (θ - \frac{1}{2} sin (2 θ)))) - \frac{1}{8} sin^{7} (θ) cos (θ)

= - \frac{1}{4} sin^{3} (θ) cos (θ) + \frac{3}{8} (θ - \frac{1}{2} sin (2 θ)) - \frac{9}{8} (- \frac{1}{6} sin^{5} (θ) cos (θ) + \frac{5}{6} (- \frac{1}{4} sin^{3} (θ) cos (θ) + \frac{3}{8} (θ - \frac{1}{2} sin (2 θ)))) - \frac{1}{8} sin^{7} (θ) cos (θ)

解答补常数

= - \frac{1}{4} sin^{3} (θ) cos (θ) + \frac{3}{8} (θ - \frac{1}{2} sin (2 θ)) - \frac{9}{8} (- \frac{1}{6} sin^{5} (θ) cos (θ) + \frac{5}{6} (- \frac{1}{4} sin^{3} (θ) cos (θ) + \frac{3}{8} (θ - \frac{1}{2} sin (2 θ)))) - \frac{1}{8} sin^{7} (θ) cos (θ) + C

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