How do you integrate #int (3x^2+x+4)/((x^2+2)(x^2+1))# using partial fractions?
The integral
By signing up, you agree to our Terms of Service and Privacy Policy
To integrate (\frac{{3x^2 + x + 4}}{{(x^2 + 2)(x^2 + 1)}}) using partial fractions, first, express the rational function as a sum of partial fractions.
[\frac{{3x^2 + x + 4}}{{(x^2 + 2)(x^2 + 1)}} = \frac{A}{x^2 + 2} + \frac{B}{x^2 + 1}]
Multiply both sides by ((x^2 + 2)(x^2 + 1)) to clear the denominators:
[3x^2 + x + 4 = A(x^2 + 1) + B(x^2 + 2)]
Expand and equate coefficients:
[3x^2 + x + 4 = Ax^2 + A + Bx^2 + 2B]
Matching coefficients:
[3 = A + B] [1 = B] [4 = A + 2B]
Solving these equations, you get (A = 1) and (B = 2).
So, (\frac{{3x^2 + x + 4}}{{(x^2 + 2)(x^2 + 1)}} = \frac{1}{{x^2 + 2}} + \frac{2}{{x^2 + 1}}).
Now, integrate each term separately:
[\int \frac{{3x^2 + x + 4}}{{(x^2 + 2)(x^2 + 1)}} dx = \int \frac{1}{{x^2 + 2}} dx + \int \frac{2}{{x^2 + 1}} dx]
[\int \frac{1}{{x^2 + 2}} dx = \frac{1}{\sqrt{2}} \arctan{\left(\frac{x}{\sqrt{2}}\right)} + C_1]
[\int \frac{2}{{x^2 + 1}} dx = 2 \arctan{x} + C_2]
Where (C_1) and (C_2) are constants of integration.
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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.

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7