What is the integral of (1/8)y_{1}e^{-(y_{1}+y_{2})/2}_{1} ?

The integral of (1/8)y_{1}e^{-(y_{1}+y_{2})/2}_{1} is -1/4 e^{(-y_{1}-y_{2})/2}(y_{1}+2)+C

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integral of (1/8)y_{1}e^{-(y_{1}+y_{2})/2}_{1}

\int (\frac{1}{8}) y_{1} e^{- \frac{y _{1} + y _{2}}{2}} d y_{1}

- \frac{1}{4} e^{\frac{- y _{1} - y _{2}}{2}} (y_{1} + 2) + C

Solution steps

\int \frac{1}{8} y_{1} e^{- \frac{y _{1} + y _{2}}{2}} d y_{1}

Take the constant out:

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

= \frac{1}{8} \cdot \int e^{- \frac{y _{2} + y _{1}}{2}} y_{1} d y_{1}

Expand

- \frac{y _{2} + y _{1}}{2} : - \frac{y _{2}}{2} - \frac{y _{1}}{2}

= \frac{1}{8} \cdot \int e^{- \frac{y _{2}}{2} - \frac{y _{1}}{2}} y_{1} d y_{1}

Simplify

e^{- \frac{y _{2}}{2} - \frac{y _{1}}{2}} : e^{- \frac{y _{2}}{2}} e^{- \frac{y _{1}}{2}}

= \frac{1}{8} \cdot \int e^{- \frac{y _{2}}{2}} e^{- \frac{y _{1}}{2}} y_{1} d y_{1}

Take the constant out:

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

= \frac{1}{8} e^{- \frac{y _{2}}{2}} \cdot \int e^{- \frac{y _{1}}{2}} y_{1} d y_{1}

Apply Integration By Parts

: - 2 e^{- \frac{y _{1}}{2}} y_{1} - \int - 2 e^{- \frac{y _{1}}{2}} d y_{1}

= \frac{1}{8} e^{- \frac{y _{2}}{2}} (- 2 e^{- \frac{y _{1}}{2}} y_{1} - \int - 2 e^{- \frac{y _{1}}{2}} d y_{1})

\int - 2 e^{- \frac{y _{1}}{2}} d y_{1} = 4 e^{- \frac{y _{1}}{2}}

= \frac{1}{8} e^{- \frac{y _{2}}{2}} (- 2 e^{- \frac{y _{1}}{2}} y_{1} - 4 e^{- \frac{y _{1}}{2}})

Simplify

\frac{1}{8} e^{- \frac{y _{2}}{2}} (- 2 e^{- \frac{y _{1}}{2}} y_{1} - 4 e^{- \frac{y _{1}}{2}}) : - \frac{1}{4} e^{\frac{- y _{1} - y _{2}}{2}} (y_{1} + 2)

= - \frac{1}{4} e^{\frac{- y _{1} - y _{2}}{2}} (y_{1} + 2)

Add a constant to the solution

= - \frac{1}{4} e^{\frac{- y _{1} - y _{2}}{2}} (y_{1} + 2) + C

f (x) = 3 sin (x) - 4 cos (x)

y^{^{'}} = 4 x y

\int_{- 2}^{1} 1 + 25 d x

x \to 27 lim (\frac{3 - 3 x}{27 - x})

f (x) = (9 - 6 x)^{2}, (1, 9)

What is the integral of (1/8)y_{1}e^{-(y_{1}+y_{2})/2}_{1} ?
The integral of (1/8)y_{1}e^{-(y_{1}+y_{2})/2}_{1} is -1/4 e^{(-y_{1}-y_{2})/2}(y_{1}+2)+C