What is the integral of (8x^3)/(sqrt(0.2x^4+8000)) ?

The integral of (8x^3)/(sqrt(0.2x^4+8000)) is 20sqrt(0.2x^4+8000)+C

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integral of (8x^3)/(sqrt(0.2x^4+8000))

\int \frac{8 x ^{3}}{0.2 x ^{4} + 8000} d x

20 0.2 x^{4} + 8000 + C

Solution steps

\int \frac{8 x ^{3}}{0.2 x ^{4} + 8000} d x

Take the constant out:

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

= 8 \cdot \int \frac{x ^{3}}{0.2 x ^{4} + 8000} d x

Apply u-substitution

: \int \frac{1}{0.8 u} d u

= 8 \cdot \int \frac{1}{0.8 u} d u

Take the constant out:

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

= 8 \cdot \frac{1}{0.8} \cdot \int \frac{1}{u} d u

Simplify

= 8 \cdot 1.25 \cdot \int \frac{1}{u} d u

Apply the Power Rule

: 2 u^{\frac{1}{2}}

= 8 \cdot 1.25 \cdot 2 u^{\frac{1}{2}}

Substitute back

u = 0.2 x^{4} + 8000

= 8 \cdot 1.25 \cdot 2 (0.2 x^{4} + 8000)^{\frac{1}{2}}

Simplify

8 \cdot 1.25 \cdot 2 (0.2 x^{4} + 8000)^{\frac{1}{2}} : 20 0.2 x^{4} + 8000

= 20 0.2 x^{4} + 8000

Add a constant to the solution

= 20 0.2 x^{4} + 8000 + C

\frac{\partial}{\partial y} (x^{y} + y)

x \to 3 lim (\frac{1}{x + 3})

f (x) = 4 x^{10} e^{x}

\int \frac{t - 2 t ^{4}}{t} d t

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

What is the integral of (8x^3)/(sqrt(0.2x^4+8000)) ?
The integral of (8x^3)/(sqrt(0.2x^4+8000)) is 20sqrt(0.2x^4+8000)+C