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受欢迎的微积分 >

integral of e^{sin^2(x)}sin(2x)

解答

\int e^{s i n^{2} (x)} sin (2 x) d x

解答

e^{s i n^{2} (x)} + C

求解步骤

\int e^{s i n^{2} (x)} sin (2 x) d x

使用三角恒等式改写

= \int e^{\frac{1 - c o s ( 2 x )}{2}} sin (2 x) d x

使用换元积分法

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

提出常数:

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

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

使用换元积分法

= \frac{1}{2} \cdot \int e^{v} \cdot 2 d v

提出常数:

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

= \frac{1}{2} \cdot 2 \cdot \int e^{v} d v

使用积分常见定则:

\int e^{v} d v = e^{v}

= \frac{1}{2} \cdot 2 e^{v}

代回

= \frac{1}{2} \cdot 2 e^{\frac{1 - c o s ( 2 x )}{2}}

化简

\frac{1}{2} \cdot 2 e^{\frac{1 - c o s ( 2 x )}{2}} : e^{s i n^{2} (x)}

= e^{s i n^{2} (x)}

解答补常数

= e^{s i n^{2} (x)} + C

作图

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\int_{\frac{1}{2}}^{1} \frac{3}{x 2 - x} d x

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