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Popular >

integral of 1/((x^2+x+1)^2)

Solution

integral

\frac{1}{( x ^{2} + x + 1 ) ^{2}}

Solution

\frac{2}{3 3} (2 arctan (\frac{2 x + 1}{3}) + sin (2 arctan (\frac{2 x + 1}{3}))) + C

Solution steps

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

Complete the square

x^{2} + x + 1 : (x + \frac{1}{2})^{2} + \frac{3}{4}

= \int \frac{1}{( ( x + \frac{1}{2} ) ^{2} + \frac{3}{4} ) ^{2}} d x

Apply u-substitution

: \int \frac{16}{( 4 u ^{2} + 3 ) ^{2}} d u

= \int \frac{16}{( 4 u ^{2} + 3 ) ^{2}} d u

Take the constant out:

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

= 16 \cdot \int \frac{1}{( 4 u ^{2} + 3 ) ^{2}} d u

Apply Integral Substitution

= 16 \cdot \int \frac{3}{2 ( 3 v ^{2} + 3 ) ^{2}} d v

Take the constant out:

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

= 16 \cdot \frac{3}{2} \cdot \int \frac{1}{( 3 v ^{2} + 3 ) ^{2}} d v

Factor

\frac{1}{( 3 v ^{2} + 3 ) ^{2}}

= 16 \cdot \frac{3}{2} \cdot \int \frac{1}{9 ( v ^{2} + 1 ) ^{2}} d v

Take the constant out:

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

= 16 \cdot \frac{3}{2} \cdot \frac{1}{9} \cdot \int \frac{1}{( v ^{2} + 1 ) ^{2}} d v

Apply Trigonometric Substitution

: \int \frac{1}{sec ^{4} ( w ) cos ^{2} ( w )} d w

= 16 \cdot \frac{3}{2} \cdot \frac{1}{9} \cdot \int \frac{1}{sec ^{4} ( w ) cos ^{2} ( w )} d w

Rewrite using trig identities

= 16 \cdot \frac{3}{2} \cdot \frac{1}{9} \cdot \int \frac{1 + cos ( 2 w )}{2} d w

Take the constant out:

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

= 16 \cdot \frac{3}{2} \cdot \frac{1}{9} \cdot \frac{1}{2} \cdot \int 1 + cos (2 w) d w

Apply the Sum Rule:

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

= 16 \cdot \frac{3}{2} \cdot \frac{1}{9} \cdot \frac{1}{2} (\int 1 d w + \int cos (2 w) d w)

\int 1 d w = w

\int cos (2 w) d w = \frac{1}{2} sin (2 w)

= 16 \cdot \frac{3}{2} \cdot \frac{1}{9} \cdot \frac{1}{2} (w + \frac{1}{2} sin (2 w))

Substitute back

= 16 \cdot \frac{3}{2} \cdot \frac{1}{9} \cdot \frac{1}{2} (arctan (\frac{2}{3} (x + \frac{1}{2})) + \frac{1}{2} sin (2 arctan (\frac{2}{3} (x + \frac{1}{2}))))

Simplify

16 \cdot \frac{3}{2} \cdot \frac{1}{9} \cdot \frac{1}{2} (arctan (\frac{2}{3} (x + \frac{1}{2})) + \frac{1}{2} sin (2 arctan (\frac{2}{3} (x + \frac{1}{2})))) : \frac{2}{3 3} (2 arctan (\frac{2 x + 1}{3}) + sin (2 arctan (\frac{2 x + 1}{3})))

= \frac{2}{3 3} (2 arctan (\frac{2 x + 1}{3}) + sin (2 arctan (\frac{2 x + 1}{3})))

Add a constant to the solution

= \frac{2}{3 3} (2 arctan (\frac{2 x + 1}{3}) + sin (2 arctan (\frac{2 x + 1}{3}))) + C

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