What is the integral of (2x^3-2x+3)^{(-1)/2}(3x^2-1) ?

The integral of (2x^3-2x+3)^{(-1)/2}(3x^2-1) is (2x^3-2x+3)^{1/2}+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (2x^3-2x+3)^{(-1)/2}(3x^2-1)

\int (2 x^{3} - 2 x + 3)^{\frac{- 1}{2}} (3 x^{2} - 1) d x

(2 x^{3} - 2 x + 3)^{\frac{1}{2}} + C

Solution steps

\int (2 x^{3} - 2 x + 3)^{\frac{- 1}{2}} (3 x^{2} - 1) d x

Apply u-substitution

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

= \int \frac{1}{2 u ^{\frac{1}{2}}} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{1}{u ^{\frac{1}{2}}} d u

Apply the Power Rule

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

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

Substitute back

u = 2 x^{3} - 2 x + 3

= \frac{1}{2} \cdot 2 (2 x^{3} - 2 x + 3)^{\frac{1}{2}}

Simplify

\frac{1}{2} \cdot 2 (2 x^{3} - 2 x + 3)^{\frac{1}{2}} : (2 x^{3} - 2 x + 3)^{\frac{1}{2}}

= (2 x^{3} - 2 x + 3)^{\frac{1}{2}}

Add a constant to the solution

= (2 x^{3} - 2 x + 3)^{\frac{1}{2}} + C

\int \frac{3 x - 2}{x ^{2} + 8 x + 25} d x

\int \frac{x}{x + x ^{3}} d x

\int \frac{1}{2 + cos ( u )} d u

\int \frac{cos ^{3} ( x )}{sin ^{3} ( x ) + 8} d x

What is the integral of (2x^3-2x+3)^{(-1)/2}(3x^2-1) ?
The integral of (2x^3-2x+3)^{(-1)/2}(3x^2-1) is (2x^3-2x+3)^{1/2}+C