What is the integral from 2 to 6 of 30x^3sqrt(1+x^2) ?

The integral from 2 to 6 of 30x^3sqrt(1+x^2) is 7844sqrt(37)-100sqrt(5)

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integral from 2 to 6 of 30x^3sqrt(1+x^2)

\int_{2}^{6} 30 x^{3} 1 + x^{2} d x

784437 - 1005

Decimal

47489.58248 \dots

Solution steps

\int_{2}^{6} 30 x^{3} 1 + x^{2} d x

Take the constant out:

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

= 30 \cdot \int_{2}^{6} x^{3} 1 + x^{2} d x

Apply u-substitution

= 30 \cdot \int_{5}^{37} \frac{( u - 1 ) u}{2} d u

Take the constant out:

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

= 30 \cdot \frac{1}{2} \cdot \int_{5}^{37} (u - 1) u d u

Expand

(u - 1) u : u^{\frac{3}{2}} - u

= 30 \cdot \frac{1}{2} \cdot \int_{5}^{37} u^{\frac{3}{2}} - u d u

Apply the Sum Rule:

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

= 30 \cdot \frac{1}{2} (\int_{5}^{37} u^{\frac{3}{2}} d u - \int_{5}^{37} u d u)

\int_{5}^{37} u^{\frac{3}{2}} d u = \frac{2738 37}{5} - 105

\int_{5}^{37} u d u = \frac{74 37 - 10 5}{3}

= 30 \cdot \frac{1}{2} (\frac{2738 37}{5} - 105 - \frac{74 37 - 10 5}{3})

Simplify

30 \cdot \frac{1}{2} (\frac{2738 37}{5} - 105 - \frac{74 37 - 10 5}{3}) : 784437 - 1005

= 784437 - 1005

What is the integral from 2 to 6 of 30x^3sqrt(1+x^2) ?
The integral from 2 to 6 of 30x^3sqrt(1+x^2) is 7844sqrt(37)-100sqrt(5)