integral of (cos(θ))^4

解答

\int (cos (θ))^{4} d θ

cos^{3} (θ) sin (θ) + \frac{3}{8} (θ - \frac{1}{4} sin (4 θ)) + C

求解步骤

\int (cos (θ))^{4} d θ

化简

= \int cos^{4} (θ) d θ

化简

cos^{4} (θ) : cos^{3} (θ) cos (θ)

= \int cos^{3} (θ) cos (θ) d θ

使用分布积分法

= cos^{3} (θ) sin (θ) - \int - 3 cos^{2} (θ) sin^{2} (θ) d θ

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

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

化简

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

解答补常数

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

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