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

The integral of (x^2)/(3+(x^3-4)^2) is 1/(3sqrt(11))arctan((x^3)/(sqrt(11)))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (x^2)/(3+(x^3-4)^2)

\int \frac{x ^{2}}{3 + ( x ^{3} - 4 ) ^{2}} d x

\frac{1}{3 11} arctan (\frac{x ^{3}}{11}) + C

Solution steps

\int \frac{x ^{2}}{3 + ( x ^{3} - 4 ) ^{2}} d x

Expand

3 + (x^{3} - 4)^{2} : x^{6} - 8 x^{3} + 19

= \int \frac{x ^{2}}{x ^{6} - 8 x ^{3} + 19} d x

Apply u-substitution

: \int \frac{1}{6 u ( u + 19 - 8 u )} d u

= \int \frac{1}{6 u ( u + 19 - 8 u )} d u

Take the constant out:

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

= \frac{1}{6} \cdot \int \frac{1}{u ( u + 19 - 8 u )} d u

Apply exponent rule:

\frac{1}{a ^{b}} = a^{- b}

\frac{1}{u ( u + 19 - 8 u )} = u^{- \frac{1}{2}} \frac{1}{( u + 19 - 8 )}

= \frac{1}{6} \cdot \int u^{- \frac{1}{2}} \frac{1}{u + 19 - 8} d u

Simplify

u^{- \frac{1}{2}} \frac{1}{u + 19 - 8} : \frac{1}{u ( u + 11 )}

= \frac{1}{6} \cdot \int \frac{1}{u ( u + 11 )} d u

Apply u-substitution

: \int \frac{2}{v ^{2} + 11} d v

= \frac{1}{6} \cdot \int \frac{2}{v ^{2} + 11} d v

Take the constant out:

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

= \frac{1}{6} \cdot 2 \cdot \int \frac{1}{v ^{2} + 11} d v

Apply Integral Substitution

= \frac{1}{6} \cdot 2 \cdot \int \frac{1}{11 ( w ^{2} + 1 )} d w

Take the constant out:

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

= \frac{1}{6} \cdot 2 \cdot \frac{1}{11} \cdot \int \frac{1}{w ^{2} + 1} d w

Use the common integral:

\int \frac{1}{w ^{2} + 1} d w = arctan (w)

= \frac{1}{6} \cdot 2 \cdot \frac{1}{11} arctan (w)

Substitute back

= \frac{1}{6} \cdot 2 \cdot \frac{1}{11} arctan (\frac{x ^{6}}{11})

Simplify

\frac{1}{6} \cdot 2 \cdot \frac{1}{11} arctan (\frac{x ^{6}}{11}) : \frac{1}{3 11} arctan (\frac{x ^{3}}{11})

= \frac{1}{3 11} arctan (\frac{x ^{3}}{11})

Add a constant to the solution

= \frac{1}{3 11} arctan (\frac{x ^{3}}{11}) + C

\int \frac{( 2 + e ^{x} )}{e ^{x}} d x

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

\int \frac{3}{y y ^{4} - 9} d y

\int 3^{x} cosh (3^{x}) d x

\int e^{c o s (y)} d y

What is the integral of (x^2)/(3+(x^3-4)^2) ?
The integral of (x^2)/(3+(x^3-4)^2) is 1/(3sqrt(11))arctan((x^3)/(sqrt(11)))+C