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

The integral from 0 to 2 of 3x(x^2+1)^3 is 234

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

234

Solution steps

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

Take the constant out:

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

= 3 \cdot \int_{0}^{2} x (x^{2} + 1)^{3} d x

Expand

x (x^{2} + 1)^{3} : x^{7} + 3 x^{5} + 3 x^{3} + x

= 3 \cdot \int_{0}^{2} x^{7} + 3 x^{5} + 3 x^{3} + x d x

Apply the Sum Rule:

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

= 3 (\int_{0}^{2} x^{7} d x + \int_{0}^{2} 3 x^{5} d x + \int_{0}^{2} 3 x^{3} d x + \int_{0}^{2} x d x)

\int_{0}^{2} x^{7} d x = 32

\int_{0}^{2} 3 x^{5} d x = 32

\int_{0}^{2} 3 x^{3} d x = 12

\int_{0}^{2} x d x = 2

= 3 (32 + 32 + 12 + 2)

Simplify

= 234

\int_{1}^{2} (4 - x^{2}) d x

\int_{- 1}^{2} x - 4 ∣ x ∣ d x

\int_{\frac{1}{x}}^{x^{2}} sin (t) d t

\int_{2}^{4} 4^{4 x} d x

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

What is the integral from 0 to 2 of 3x(x^2+1)^3 ?
The integral from 0 to 2 of 3x(x^2+1)^3 is 234