What is the integral from 0 to y+z of 1/((z+y+x)^3) ?

The integral from 0 to y+z of 1/((z+y+x)^3) is 3/(2(2z+2y)^2)

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integral from 0 to y+z of 1/((z+y+x)^3)

\int_{0}^{y + z} \frac{1}{( z + y + x ) ^{3}} d x

\frac{3}{2 ( 2 z + 2 y ) ^{2}}

Solution steps

\int_{0}^{y + z} \frac{1}{( z + y + x ) ^{3}} d x

Apply u-substitution

: \int_{z + y}^{2 z + 2 y} \frac{1}{u ^{3}} d u

= \int_{z + y}^{2 z + 2 y} \frac{1}{u ^{3}} d u

Apply the Power Rule

: [- \frac{1}{2 u ^{2}}]_{z + y}^{2 z + 2 y}

= [- \frac{1}{2 u ^{2}}]_{z + y}^{2 z + 2 y}

Compute the boundaries:

- \frac{1}{2 ( 2 z + 2 y ) ^{2}} + \frac{1}{2 ( z + y ) ^{2}}

= - \frac{1}{2 ( 2 z + 2 y ) ^{2}} + \frac{1}{2 ( z + y ) ^{2}}

Simplify

= \frac{3}{2 ( 2 z + 2 y ) ^{2}}

y^{^{'}} = \frac{1}{x} y - (\frac{x ^{2}}{3}) y^{4}

x + 2 y = 2, y - x = 1, 2 x + y = 7

\frac{d}{d x} (\frac{π}{x})

5 cos (π t)

\int_{0}^{3} (x + 1) (2 x - 3) d x

What is the integral from 0 to y+z of 1/((z+y+x)^3) ?
The integral from 0 to y+z of 1/((z+y+x)^3) is 3/(2(2z+2y)^2)