What is the integral of 7xe^{-x(y+7)} ?

The integral of 7xe^{-x(y+7)} is 7/((-7-y)^2)(-e^{-x(7+y)}x(7+y)-e^{-x(7+y)})+C

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integral of 7xe^{-x(y+7)}

\int 7 x e^{- x (y + 7)} d x

\frac{7}{( - 7 - y ) ^{2}} (- e^{- x (7 + y)} x (7 + y) - e^{- x (7 + y)}) + C

Solution steps

\int 7 x e^{- x (y + 7)} d x

Take the constant out:

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

= 7 \cdot \int e^{- (7 + y) x} x d x

Apply u-substitution

: \int \frac{e ^{u} u}{( - 7 - y ) ^{2}} d u

= 7 \cdot \int \frac{e ^{u} u}{( - 7 - y ) ^{2}} d u

Take the constant out:

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

= 7 \cdot \frac{1}{( - 7 - y ) ^{2}} \cdot \int e^{u} u d u

Apply Integration By Parts

: e^{u} u - \int e^{u} d u

= 7 \cdot \frac{1}{( - 7 - y ) ^{2}} (e^{u} u - \int e^{u} d u)

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

= 7 \cdot \frac{1}{( - 7 - y ) ^{2}} (e^{u} u - e^{u})

Substitute back

u = - (7 + y) x

= 7 \cdot \frac{1}{( - 7 - y ) ^{2}} (e^{- (7 + y) x} (- (7 + y) x) - e^{- (7 + y) x})

Simplify

7 \cdot \frac{1}{( - 7 - y ) ^{2}} (e^{- (7 + y) x} (- (7 + y) x) - e^{- (7 + y) x}) : \frac{7}{( - 7 - y ) ^{2}} (- e^{- x (7 + y)} x (7 + y) - e^{- x (7 + y)})

= \frac{7}{( - 7 - y ) ^{2}} (- e^{- x (7 + y)} x (7 + y) - e^{- x (7 + y)})

Add a constant to the solution

= \frac{7}{( - 7 - y ) ^{2}} (- e^{- x (7 + y)} x (7 + y) - e^{- x (7 + y)}) + C

f (x) = ln (x x^{2} + 1)

f (x) = 3 x - 6 x

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What is the integral of 7xe^{-x(y+7)} ?
The integral of 7xe^{-x(y+7)} is 7/((-7-y)^2)(-e^{-x(7+y)}x(7+y)-e^{-x(7+y)})+C