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Popular >

integral of 1/((x^2+3x+5)^3)

Solution

\int \frac{1}{( x ^{2} + 3 x + 5 ) ^{3}} d x

Solution

\frac{3 ( arctan ( \frac{2}{11} ( x + \frac{3}{2} ) ) ( 4 x ^{2} + 12 x + 20 ) + 11 ( 2 x + 3 ) )}{121 11 ( x ^{2} + 3 x + 5 )} + \frac{2 x + 3}{22 ( x ^{2} + 3 x + 5 ) ^{2}} + C

Solution steps

\int \frac{1}{( x ^{2} + 3 x + 5 ) ^{3}} d x

Complete the square

x^{2} + 3 x + 5 : (x + \frac{3}{2})^{2} + \frac{11}{4}

= \int \frac{1}{( ( x + \frac{3}{2} ) ^{2} + \frac{11}{4} ) ^{3}} d x

Apply u-substitution

: \int \frac{64}{( 4 u ^{2} + 11 ) ^{3}} d u

= \int \frac{64}{( 4 u ^{2} + 11 ) ^{3}} d u

Take the constant out:

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

= 64 \cdot \int \frac{1}{( 4 u ^{2} + 11 ) ^{3}} d u

Apply Integral Substitution

= 64 \cdot \int \frac{11}{2 ( 11 v ^{2} + 11 ) ^{3}} d v

Take the constant out:

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

= 64 \cdot \frac{11}{2} \cdot \int \frac{1}{( 11 v ^{2} + 11 ) ^{3}} d v

Factor

\frac{1}{( 11 v ^{2} + 11 ) ^{3}}

= 64 \cdot \frac{11}{2} \cdot \int \frac{1}{1331 ( v ^{2} + 1 ) ^{3}} d v

Take the constant out:

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

= 64 \cdot \frac{11}{2} \cdot \frac{1}{1331} \cdot \int \frac{1}{( v ^{2} + 1 ) ^{3}} d v

Apply Integral Reduction

= 64 \cdot \frac{11}{2} \cdot \frac{1}{1331} (\frac{3}{4} \cdot \int \frac{1}{( v ^{2} + 1 ) ^{2}} d v + \frac{v}{2 \cdot 2 ( v ^{2} + 1 ) ^{2}})

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

= 64 \cdot \frac{11}{2} \cdot \frac{1}{1331} (\frac{3}{4} \cdot \frac{1}{2} (arctan (v) + \frac{v}{1 + v ^{2}}) + \frac{v}{2 \cdot 2 ( v ^{2} + 1 ) ^{2}})

Substitute back

= 64 \cdot \frac{11}{2} \cdot \frac{1}{1331} \frac{3}{4} \cdot \frac{1}{2} arctan (\frac{2}{11} (x + \frac{3}{2})) + \frac{\frac{2}{11} ( x + \frac{3}{2} )}{1 + ( \frac{2}{11} ( x + \frac{3}{2} ) ) ^{2}} + \frac{\frac{2}{11} ( x + \frac{3}{2} )}{2 \cdot 2 ( ( \frac{2}{11} ( x + \frac{3}{2} ) ) ^{2} + 1 ) ^{2}}

Simplify

64 \cdot \frac{11}{2} \cdot \frac{1}{1331} \frac{3}{4} \cdot \frac{1}{2} arctan (\frac{2}{11} (x + \frac{3}{2})) + \frac{\frac{2}{11} ( x + \frac{3}{2} )}{1 + ( \frac{2}{11} ( x + \frac{3}{2} ) ) ^{2}} + \frac{\frac{2}{11} ( x + \frac{3}{2} )}{2 \cdot 2 ( ( \frac{2}{11} ( x + \frac{3}{2} ) ) ^{2} + 1 ) ^{2}} : \frac{3 ( arctan ( \frac{2}{11} ( x + \frac{3}{2} ) ) ( 4 x ^{2} + 12 x + 20 ) + 11 ( 2 x + 3 ) )}{121 11 ( x ^{2} + 3 x + 5 )} + \frac{2 x + 3}{22 ( x ^{2} + 3 x + 5 ) ^{2}}

= \frac{3 ( arctan ( \frac{2}{11} ( x + \frac{3}{2} ) ) ( 4 x ^{2} + 12 x + 20 ) + 11 ( 2 x + 3 ) )}{121 11 ( x ^{2} + 3 x + 5 )} + \frac{2 x + 3}{22 ( x ^{2} + 3 x + 5 ) ^{2}}

Add a constant to the solution

= \frac{3 ( arctan ( \frac{2}{11} ( x + \frac{3}{2} ) ) ( 4 x ^{2} + 12 x + 20 ) + 11 ( 2 x + 3 ) )}{121 11 ( x ^{2} + 3 x + 5 )} + \frac{2 x + 3}{22 ( x ^{2} + 3 x + 5 ) ^{2}} + C

Graph

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