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受欢迎的微积分 >

integral of ((5sin^3(x)))/(cos(x))

解答

\int \frac{( 5 sin ^{3} ( x ) )}{cos ( x )} d x

解答

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

求解步骤

\int \frac{5 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

= 5 \cdot \int \frac{sin ^{3} ( x )}{cos ( x )} d x

使用三角恒等式改写

= 5 \cdot \int \frac{( 1 - cos ^{2} ( x ) ) sin ( x )}{cos ( x )} d x

使用换元积分法

= 5 \cdot \int - \frac{1 - u ^{2}}{u} d u

乘开

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

= 5 \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

= 5 (- \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}

= 5 (- ln ∣ u ∣ + \frac{u ^{2}}{2})

u = cos (x)

代回

= 5 (- ln ∣ cos (x) ∣ + \frac{cos ^{2} ( x )}{2})

解答补常数

= 5 (- ln ∣ cos (x) ∣ + \frac{cos ^{2} ( x )}{2}) + C

作图

流行的例子

integral of 5e^{9x}

\int 5 e^{9 x} d x

tangent f(x)=x^4,(2,16)

t an g e n t f (x) = x^{4}, (2, 16)

limit as x approaches 0 of+(x)

x \to 0 lim (+ (x))

integral of 1/((x/2)^2+1)

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

derivative of \sqrt[3]{8}

\frac{d}{d x} (38)