What is the integral from y/2 to sqrt(y of)(4x+2) ?

The integral from y/2 to sqrt(y of)(4x+2) is 2sqrt(y)+y-(y^2)/2

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integral from y/2 to sqrt(y of)(4x+2)

\int_{\frac{y}{2}}^{y} (4 x + 2) d x

2 y + y - \frac{y ^{2}}{2}

Solution steps

\int_{\frac{y}{2}}^{y} 4 x + 2 d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{\frac{y}{2}}^{y} 2 d x + \int_{\frac{y}{2}}^{y} 4 x d x

\int_{\frac{y}{2}}^{y} 2 d x = 2 y - y

\int_{\frac{y}{2}}^{y} 4 x d x = 4 (\frac{y}{2} - \frac{y ^{2}}{8})

= 2 y - y + 4 (\frac{y}{2} - \frac{y ^{2}}{8})

Simplify

2 y - y + 4 (\frac{y}{2} - \frac{y ^{2}}{8}) : 2 y + y - \frac{y ^{2}}{2}

= 2 y + y - \frac{y ^{2}}{2}

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What is the integral from y/2 to sqrt(y of)(4x+2) ?
The integral from y/2 to sqrt(y of)(4x+2) is 2sqrt(y)+y-(y^2)/2