What is the integral from-3 to-1 of y^3sqrt(49y^2-4) ?

The integral from-3 to-1 of y^3sqrt(49y^2-4) is -(581647sqrt(437)-20925sqrt(5))/(36015)

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integral from-3 to-1 of y^3sqrt(49y^2-4)

\int_{- 3}^{- 1} y^{3} 49 y^{2} - 4 d y

- \frac{581647 437 - 20925 5}{36015}

Decimal

- 336.31198 \dots

Solution steps

\int_{- 3}^{- 1} y^{3} 49 y^{2} - 4 d y

Apply u-substitution

= \int_{- 27}^{- 1} \frac{u ^{\frac{1}{3}} 49 u ^{\frac{2}{3}} - 4}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int_{- 27}^{- 1} u^{\frac{1}{3}} 49 u^{\frac{2}{3}} - 4 d u

Apply u-substitution

Take the constant out:

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

Apply u-substitution

= \frac{1}{3} \cdot 3 \cdot \int_{441 (- 1)^{\frac{2}{3}} - 4}^{49 (- 1)^{\frac{2}{3}} - 4} \frac{w ( w + 4 )}{4802} d w

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= \frac{1}{3} \cdot 3 - \int_{49 (- 1)^{\frac{2}{3}} - 4}^{441 (- 1)^{\frac{2}{3}} - 4} \frac{w ( w + 4 )}{4802} d w

Take the constant out:

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

= \frac{1}{3} \cdot 3 - \frac{1}{4802} \cdot \int_{49 (- 1)^{\frac{2}{3}} - 4}^{441 (- 1)^{\frac{2}{3}} - 4} w (w + 4) d w

Expand

w (w + 4) : w^{\frac{3}{2}} + 4 w

= \frac{1}{3} \cdot 3 - \frac{1}{4802} \cdot \int_{49 (- 1)^{\frac{2}{3}} - 4}^{441 (- 1)^{\frac{2}{3}} - 4} w^{\frac{3}{2}} + 4 w d w

Apply the Sum Rule:

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

= \frac{1}{3} \cdot 3 - \frac{1}{4802} \int_{49 (- 1)^{\frac{2}{3}} - 4}^{441 (- 1)^{\frac{2}{3}} - 4} w^{\frac{3}{2}} d w + \int_{49 (- 1)^{\frac{2}{3}} - 4}^{441 (- 1)^{\frac{2}{3}} - 4} 4 w d w

\int_{49 (- 1)^{\frac{2}{3}} - 4}^{441 (- 1)^{\frac{2}{3}} - 4} w^{\frac{3}{2}} d w = \frac{381938 437}{5} - 24305

\int_{49 (- 1)^{\frac{2}{3}} - 4}^{441 (- 1)^{\frac{2}{3}} - 4} 4 w d w = 4 (\frac{874 437}{3} - 905)

= \frac{1}{3} \cdot 3 (- \frac{1}{4802} (\frac{381938 437}{5} - 24305 + 4 (\frac{874 437}{3} - 905)))

Simplify

\frac{1}{3} \cdot 3 (- \frac{1}{4802} (\frac{381938 437}{5} - 24305 + 4 (\frac{874 437}{3} - 905))) : - \frac{581647 437 - 20925 5}{36015}

= - \frac{581647 437 - 20925 5}{36015}

What is the integral from-3 to-1 of y^3sqrt(49y^2-4) ?
The integral from-3 to-1 of y^3sqrt(49y^2-4) is -(581647sqrt(437)-20925sqrt(5))/(36015)