What is the integral from 1 to 8 of 2pi(y+3)y^{-2} ?

The integral from 1 to 8 of 2pi(y+3)y^{-2} is 2pi(3ln(2)+21/8)

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integral from 1 to 8 of 2pi(y+3)y^{-2}

\int_{1}^{8} 2 π (y + 3) y^{- 2} d y

2 π (3 ln (2) + \frac{21}{8})

Decimal Notation

29.55887 \dots

Solution steps

\int_{1}^{8} 2 π (y + 3) y^{- 2} d y

Take the constant out:

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

= 2 π \cdot \int_{1}^{8} (y + 3) y^{- 2} d y

Simplify

(y + 3) y^{- 2} : \frac{y + 3}{y ^{2}}

= 2 π \cdot \int_{1}^{8} \frac{y + 3}{y ^{2}} d y

Expand

\frac{y + 3}{y ^{2}} : \frac{1}{y} + \frac{3}{y ^{2}}

= 2 π \cdot \int_{1}^{8} \frac{1}{y} + \frac{3}{y ^{2}} d y

Apply the Sum Rule:

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

= 2 π (\int_{1}^{8} \frac{1}{y} d y + \int_{1}^{8} \frac{3}{y ^{2}} d y)

\int_{1}^{8} \frac{1}{y} d y = 3 ln (2)

\int_{1}^{8} \frac{3}{y ^{2}} d y = \frac{21}{8}

= 2 π (3 ln (2) + \frac{21}{8})

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What is the integral from 1 to 8 of 2pi(y+3)y^{-2} ?
The integral from 1 to 8 of 2pi(y+3)y^{-2} is 2pi(3ln(2)+21/8)