What is the integral of x/(sqrt(1-3x^2-2x^4)) ?

The integral of x/(sqrt(1-3x^2-2x^4)) is 1/(2sqrt(2))arcsin((4x^2+3)/(sqrt(17)))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of x/(sqrt(1-3x^2-2x^4))

\int \frac{x}{1 - 3 x ^{2} - 2 x ^{4}} d x

\frac{1}{2 2} arcsin (\frac{4 x ^{2} + 3}{17}) + C

Solution steps

\int \frac{x}{1 - 3 x ^{2} - 2 x ^{4}} d x

Apply u-substitution

: \int \frac{1}{2 1 - 3 u - 2 u ^{2}} d u

= \int \frac{1}{2 1 - 3 u - 2 u ^{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}{1 - 3 u - 2 u ^{2}} d u

Complete the square

1 - 3 u - 2 u^{2} : - 2 (u + \frac{3}{4})^{2} + \frac{17}{8}

= \frac{1}{2} \cdot \int \frac{1}{- 2 ( u + \frac{3}{4} ) ^{2} + \frac{17}{8}} d u

Apply u-substitution

: \int \frac{2 2}{- 16 v ^{2} + 17} d v

= \frac{1}{2} \cdot \int \frac{2 2}{- 16 v ^{2} + 17} d v

Take the constant out:

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

= \frac{1}{2} \cdot 22 \cdot \int \frac{1}{- 16 v ^{2} + 17} d v

Apply Trigonometric Substitution

: \int \frac{1}{4} d w

= \frac{1}{2} \cdot 22 \cdot \int \frac{1}{4} d w

Take the constant out:

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

= \frac{1}{2} \cdot 22 \frac{1}{4} \cdot \int 1 d w

Integral of a constant:

\int a d x = a x

= \frac{1}{2} \cdot 22 \frac{1}{4} \cdot 1 \cdot w

Substitute back

= \frac{1}{2} \cdot 22 \frac{1}{4} \cdot 1 \cdot arcsin (\frac{4}{17} (x^{2} + \frac{3}{4}))

Simplify

\frac{1}{2} \cdot 22 \frac{1}{4} \cdot 1 \cdot arcsin (\frac{4}{17} (x^{2} + \frac{3}{4})) : \frac{1}{2 2} arcsin (\frac{4 x ^{2} + 3}{17})

= \frac{1}{2 2} arcsin (\frac{4 x ^{2} + 3}{17})

Add a constant to the solution

= \frac{1}{2 2} arcsin (\frac{4 x ^{2} + 3}{17}) + C

\int a^{2} + b^{2} x^{2} (x) d x

\int e^{50 x} d x

\int 4 x^{3} e^{x^{4} + 1} d x

\int \frac{( x - 1 ) ^{2}}{81} d x

\int \frac{e ^{x}}{9 + 4 e ^{2 x}} d x

What is the integral of x/(sqrt(1-3x^2-2x^4)) ?
The integral of x/(sqrt(1-3x^2-2x^4)) is 1/(2sqrt(2))arcsin((4x^2+3)/(sqrt(17)))+C