integral of 3ye^{y^3}

解答

\int 3 y e^{y^{3}} d y

3 \cdot \frac{1}{2} - \frac{y ^{2} Γ ( \frac{1}{\frac{3}{2}} , - ( y ^{2} ) ^{\frac{3}{2}} )}{\frac{3}{2} \frac{3}{2} - ( y ^{2} ) ^{\frac{3}{2}}} + C

求解步骤

\int 3 y e^{y^{3}} d y

提出常数:

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

= 3 \cdot \int y e^{y^{3}} d y

使用换元积分法

= 3 \cdot \int \frac{e ^{u^{\frac{3}{2}}}}{2} d u

提出常数:

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

= 3 \cdot \frac{1}{2} \cdot \int e^{u^{\frac{3}{2}}} d u

这是非初等函数积分:

\int e^{x^{a}} d x = - \frac{x Γ ( \frac{1}{a} , - x ^{a} )}{a a - x ^{a}}

= 3 \cdot \frac{1}{2} - \frac{u Γ ( \frac{1}{\frac{3}{2}} , - u ^{\frac{3}{2}} )}{\frac{3}{2} \frac{3}{2} - u ^{\frac{3}{2}}}

u = y^{2}

代回

= 3 \cdot \frac{1}{2} - \frac{y ^{2} Γ ( \frac{1}{\frac{3}{2}} , - ( y ^{2} ) ^{\frac{3}{2}} )}{\frac{3}{2} \frac{3}{2} - ( y ^{2} ) ^{\frac{3}{2}}}

解答补常数

= 3 \cdot \frac{1}{2} - \frac{y ^{2} Γ ( \frac{1}{\frac{3}{2}} , - ( y ^{2} ) ^{\frac{3}{2}} )}{\frac{3}{2} \frac{3}{2} - ( y ^{2} ) ^{\frac{3}{2}}} + C

f (x) = ln (x - 4)

\int - cos^{2} (t) d t

\frac{\partial}{\partial x} (2^{x^{2} - y^{5}})

t an g e n t x^{3} - 8 x^{2} + 5 x - 2

\frac{\partial}{\partial x} (4 e^{x y + 3})