What is the integral from 0 to 3 of x^5sqrt(2x^3+1) ?

The integral from 0 to 3 of x^5sqrt(2x^3+1) is (4400sqrt(55)+1)/(45)

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integral from 0 to 3 of x^5sqrt(2x^3+1)

\int_{0}^{3} x^{5} 2 x^{3} + 1 d x

\frac{4400 55 + 1}{45}

Decimal

725.16162 \dots

Solution steps

\int_{0}^{3} x^{5} 2 x^{3} + 1 d x

Apply u-substitution

: \int_{1}^{55} \frac{u ( u - 1 )}{12} d u

= \int_{1}^{55} \frac{u ( u - 1 )}{12} d u

Take the constant out:

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

= \frac{1}{12} \cdot \int_{1}^{55} u (u - 1) d u

Expand

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

= \frac{1}{12} \cdot \int_{1}^{55} 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

= \frac{1}{12} (\int_{1}^{55} u^{\frac{3}{2}} d u - \int_{1}^{55} u d u)

\int_{1}^{55} u^{\frac{3}{2}} d u = 121055 - \frac{2}{5}

\int_{1}^{55} u d u = \frac{110 55 - 2}{3}

= \frac{1}{12} (121055 - \frac{2}{5} - \frac{110 55 - 2}{3})

Simplify

\frac{1}{12} (121055 - \frac{2}{5} - \frac{110 55 - 2}{3}) : \frac{4400 55 + 1}{45}

= \frac{4400 55 + 1}{45}

\frac{d}{d x} (5^{2 x + 1})

y^{^{''}} - 2 y^{^{'}} + y = \frac{( e ^{t} )}{1 + t ^{2}}

x \to 0 - lim (12 e^{- \frac{2}{x}})

f (x) = \frac{x ^{2} - 1}{x ^{2} + x + 1}

x \to - 2 lim (x^{2} - x + 4)

What is the integral from 0 to 3 of x^5sqrt(2x^3+1) ?
The integral from 0 to 3 of x^5sqrt(2x^3+1) is (4400sqrt(55)+1)/(45)