How do you integrate #int 1/((x+1)(x^2+2x+2))# using partial fractions?
The answer is
The decomposition in partial fractions will be discussed first.
#C=1
In order to get the second integral, we substitute
When everything is combined,
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To integrate ( \frac{1}{(x+1)(x^2 + 2x + 2)} ) using partial fractions, follow these steps:

Decompose the fraction into partial fractions: ( \frac{1}{(x+1)(x^2 + 2x + 2)} = \frac{A}{x+1} + \frac{Bx + C}{x^2 + 2x + 2} )

Clear the denominator by multiplying both sides by ( (x+1)(x^2 + 2x + 2) ): ( 1 = A(x^2 + 2x + 2) + (Bx + C)(x + 1) )

Expand and group like terms: ( 1 = Ax^2 + 2Ax + 2A + Bx^2 + Bx + Cx + C )

Equate coefficients of like terms: ( A + B = 0 ) (for ( x^2 ) terms)
( 2A + B + C = 0 ) (for ( x ) terms)
( 2A + C = 1 ) (for constant terms) 
Solve the system of equations to find the values of ( A ), ( B ), and ( C ).

Once you have found the values of ( A ), ( B ), and ( C ), rewrite the original fraction with the partial fractions:
( \frac{1}{(x+1)(x^2 + 2x + 2)} = \frac{A}{x+1} + \frac{Bx + C}{x^2 + 2x + 2} )

Now integrate each partial fraction separately.
( \int \frac{1}{(x+1)(x^2 + 2x + 2)} , dx = \int \frac{A}{x+1} , dx + \int \frac{Bx + C}{x^2 + 2x + 2} , dx )
( = A \lnx+1 + \int \frac{Bx + C}{x^2 + 2x + 2} , dx )

For the second integral, complete the square in the denominator, then use a substitution method to integrate.

After integrating both parts, you'll have your final result.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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