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受欢迎的微积分 >

integral of (7sin^3(x))/(cos(x))

解答

\int \frac{7 sin ^{3} ( x )}{cos ( x )} d x

解答

7 (- ln ∣ cos (x) ∣ + \frac{cos ^{2} ( x )}{2}) + C

求解步骤

\int \frac{7 sin ^{3} ( x )}{cos ( x )} d x

提出常数:

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

= 7 \cdot \int \frac{sin ^{3} ( x )}{cos ( x )} d x

使用三角恒等式改写

= 7 \cdot \int \frac{( 1 - cos ^{2} ( x ) ) sin ( x )}{cos ( x )} d x

使用换元积分法

= 7 \cdot \int - \frac{1 - u ^{2}}{u} d u

乘开

- \frac{1 - u ^{2}}{u} : - \frac{1}{u} + u

= 7 \cdot \int - \frac{1}{u} + u d u

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

= 7 (- \int \frac{1}{u} d u + \int u d u)

\int \frac{1}{u} d u = ln ∣ u ∣

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

= 7 (- ln ∣ u ∣ + \frac{u ^{2}}{2})

u = cos (x)

代回

= 7 (- ln ∣ cos (x) ∣ + \frac{cos ^{2} ( x )}{2})

解答补常数

= 7 (- ln ∣ cos (x) ∣ + \frac{cos ^{2} ( x )}{2}) + C

作图

流行的例子

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\int \frac{1}{x \cdot ( 1 + ln ^{2} ( x ) )} d x

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\frac{d}{d x} (sin^{2} (a x))

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