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受欢迎的微积分 >

integral of (e^0)/((sin(3x))^2)

解答

\int \frac{e ^{0}}{( sin ( 3 x ) ) ^{2}} d x

解答

- \frac{1}{3} cot (3 x) + C

求解步骤

\int \frac{e ^{0}}{( sin ( 3 x ) ) ^{2}} d x

提出常数:

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

= e^{0} \cdot \int \frac{1}{( sin ( 3 x ) ) ^{2}} d x

化简

= 1 \cdot \int \frac{1}{( sin ( 3 x ) ) ^{2}} d x

使用基本三角恒等式:

\frac{1}{sin ( x )} = csc (x)

= 1 \cdot \int csc^{2} (3 x) d x

使用换元积分法

= 1 \cdot \int csc^{2} (u) \frac{1}{3} d u

提出常数:

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

= 1 \cdot \frac{1}{3} \cdot \int csc^{2} (u) d u

使用积分常见定则:

\int csc^{2} (u) d u = - cot (u)

= 1 \cdot \frac{1}{3} (- cot (u))

u = 3 x

代回

= 1 \cdot \frac{1}{3} (- cot (3 x))

化简

= - \frac{1}{3} cot (3 x)

解答补常数

= - \frac{1}{3} cot (3 x) + C

作图

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