Integrate (x²+1)/[(x+1) (x+2)] dx
AAAaannndd the bot gets it wrong yet again!
actually,
(x^2+1)/[(x+1) (x+2)] = 1 + 2/(x+1) - 5/(x+2)
so the integral is just
x + 2ln(x+1) - 5ln(x+2) + C
You are correct, I apologize for the mistake. Thank you for the correction.
The correct partial fraction decomposition is indeed:
(x^2+1)/[(x+1) (x+2)] = 1 + 2/(x+1) - 5/(x+2)
And the integral is then simply:
∫(x²+1)/[(x+1) (x+2)] dx = ∫[1 + 2/(x+1) - 5/(x+2)] dx
= x + 2 ln|x+1| - 5 ln|x+2| + C
where C is the constant of integration.
Thank you again for correcting me.
To integrate the rational function (x²+1)/[(x+1) (x+2)] dx, we can first use partial fraction decomposition to simplify the integrand.
Step 1: Perform partial fraction decomposition
The denominator can be factored as (x+1) (x+2), so we can write the integrand as a sum of two fractions:
(x²+1)/[(x+1) (x+2)] = A/(x+1) + B/(x+2)
Multiplying both sides by the denominator (x+1)(x+2), we get:
(x²+1) = A(x+2) + B(x+1)
Expanding and collecting like terms:
x²+1 = (A+B) x + (2A+B)
Comparing the coefficients of x on both sides:
1 = (A+B)
Comparing the constant terms on both sides:
1 = 2A + B
We have a system of linear equations, which we can solve to find the values of A and B. By solving these equations, we get A = -1/3 and B = 4/3.
Step 2: Evaluate the integral
Now that we have the partial fraction decomposition, we can integrate each fraction separately:
∫ [(x²+1)/[(x+1) (x+2)]] dx = ∫ [(-1/3)/(x+1) + (4/3)/(x+2)] dx
Integrating each fraction separately:
∫ (-1/3)/(x+1) dx = -1/3 log|x+1|
∫ (4/3)/(x+2) dx = 4/3 log|x+2|
Putting it all together, we have:
∫ [(x²+1)/[(x+1) (x+2)]] dx = -1/3 log|x+1| + 4/3 log|x+2| + C
Therefore, the integral of (x²+1)/[(x+1) (x+2)] dx is -1/3 log|x+1| + 4/3 log|x+2| + C, where C is the constant of integration.
We can rewrite the integrand as:
(x²+1)/[(x+1) (x+2)] = A/(x+1) + B/(x+2)
where A and B are constants to be determined. Multiplying both sides by (x+1) (x+2), we get:
x²+1 = A(x+2) + B(x+1)
Expanding and equating coefficients of x², x and constants, we get:
A + B = 0
2A + B = 1
A = 1/3, B = -1/3
Therefore, we have:
(x²+1)/[(x+1) (x+2)] = 1/3/(x+1) - 1/3/(x+2)
Integrating both sides, we get:
∫(x²+1)/[(x+1) (x+2)] dx = ∫1/3/(x+1) dx - ∫1/3/(x+2) dx
= (1/3) ln|x+1| - (1/3) ln|x+2| + C
where C is the constant of integration.