How do you integrate #int (3x^2+x+4)/((x^2+2)(x^2+1))# using partial fractions?

Answer 1

The integral #=-ln(x^2+2)/2+ln(x^2+1)/2+arctanx+sqrt2arctan(x/sqrt2)+C#

The decomposition in partial fractions is #(3x^2+x+4)/((x^2+2)(x^2+1))=(Ax+B)/(x^2+2)+(Cx+D)/(x^2+1)#
#3x^2+x+4=(Ax+B)(x^2+1)+(Cx+D)(x^2+2)# Let #x=0# then 4= B+2D Coefficients of #x^2#, #3=B+D# From thes equations. we get #D=1# and #B=2# So, #3x^2+x+4=(Ax+2)(x^2+1)+(Cx+1)(x^2+2)# We compare the coefficients of #x# #1=A+2C# and coefficients of #X^3#, #0=A+C# So, #A=-1# and #C=1#
#(3x^2+x+4)/((x^2+2)(x^2+1))=(-x+2)/(x^2+2)+(x+1)/(x^2+1)# #int((3x^2+x+4)dx)/((x^2+2)(x^2+1))=int((-x+2)dx)/(x^2+2)+int((x+1)dx)/(x^2+1)# #=int(-xdx)/(x^2+2)+int(2dx)/(x^2+2)+int(xdx)/(x^2+1)+intdx/(x^2+1)# #=-ln(x^2+2)/2+ln(x^2+1)/2+arctanx+sqrt2arctan(x/sqrt2)+C# as#intdx/(x^2+1)=arctanx#
and #int(xdx)/(x^2+1)=ln(x^2+1)/2#
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Answer 2

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.

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Answer from HIX Tutor

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.

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