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受欢迎的微积分 >

integral from pi/4 to pi/2 of 4cos^4(x)

解答

\int_{\frac{π}{4}}^{\frac{π}{2}} 4 cos^{4} (x) d x

解答

\frac{3 π - 8}{8}

+1

十进制

0.17809 \dots

求解步骤

\int_{\frac{π}{4}}^{\frac{π}{2}} 4 cos^{4} (x) d x

提出常数:

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

= 4 \cdot \int_{\frac{π}{4}}^{\frac{π}{2}} cos^{4} (x) d x

化简

cos^{4} (x) : cos^{3} (x) cos (x)

= 4 \cdot \int_{\frac{π}{4}}^{\frac{π}{2}} cos^{3} (x) cos (x) d x

使用分布积分法

= 4 [cos^{3} (x) sin (x) - \int - 3 cos^{2} (x) sin^{2} (x) d x]_{\frac{π}{4}}^{\frac{π}{2}}

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

= 4 [cos^{3} (x) sin (x) - (- \frac{3}{8} (x - \frac{1}{4} sin (4 x)))]_{\frac{π}{4}}^{\frac{π}{2}}

化简

4 [cos^{3} (x) sin (x) - (- \frac{3}{8} (x - \frac{1}{4} sin (4 x)))]_{\frac{π}{4}}^{\frac{π}{2}} : 4 [\frac{1}{32} (32 cos^{3} (x) sin (x) + 3 (4 x - sin (4 x)))]_{\frac{π}{4}}^{\frac{π}{2}}

= 4 [\frac{1}{32} (32 cos^{3} (x) sin (x) + 3 (4 x - sin (4 x)))]_{\frac{π}{4}}^{\frac{π}{2}}

带入上下限计算后得:

\frac{3 π - 8}{32}

= 4 \cdot \frac{3 π - 8}{32}

化简

= \frac{3 π - 8}{8}

作图

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