Evaluate the integral. ∫[0,1](12/4x^2+5x+1)dx
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To evaluate the given integral ∫[0,1](12/4x^2+5x+1)dx, follow these steps:
Step 1: Simplify the integrand
The integrand can be simplified by factoring the denominator. Therefore, we have:
∫[0,1](12/(4x^2+5x+1))dx
Step 2: Factor the denominator
To factor the denominator, we need to find two numbers whose sum is 5 and whose product is 4 multiplied by 1. In this case, the numbers are 1 and 4:
4x^2 + 5x + 1 = (4x + 1)(x + 1)
Step 3: Perform a partial fraction decomposition
To integrate the simplified function, we use partial fraction decomposition. Let's express the integrand as the sum of two fractions with appropriate constants:
12/(4x^2 + 5x + 1) = A/(4x + 1) + B/(x + 1)
To find the constants A and B, we need to cross-multiply and equate coefficients:
12 = A(x + 1) + B(4x + 1)
Expanding and matching the coefficients, we get:
12 = (A + 4B)x + (A + B)
Matching the coefficients of the constant term and the x term, we get the following system of equations:
A + B = 12 ...(1)
A + 4B = 0 ...(2)
Solving the system of equations, we find:
A = 12/3 = 4
B = -12/3 = -4
So, the partial fraction decomposition becomes:
12/(4x^2 + 5x + 1) = 4/(4x + 1) - 4/(x + 1)
Step 4: Integrate each term separately
Now we can integrate each term separately. The integral of 4/(4x + 1) can be evaluated using the substitution method, where u = 4x + 1:
∫(4/(4x + 1))dx = ∫(4/u)du = 4ln|u| + C = 4ln|4x + 1| + C1
The integral of -4/(x + 1) can be evaluated by using the natural logarithm property:
∫(-4/(x + 1))dx = -4ln|x + 1| + C = -4ln|1 + x| + C2
Step 5: Apply the limits of integration
To find the definite integral from 0 to 1, we substitute the upper limit and lower limit separately into the antiderivatives obtained in Step 4:
∫[0,1](12/(4x^2 + 5x + 1))dx = [4ln|4x + 1| - 4ln|1 + x|] evaluated from 0 to 1
Substituting the upper limit 1 into the antiderivatives:
[4ln|4(1) + 1| - 4ln|1 + 1|]
Simplifying:
[4ln|5| - 4ln|2|]
Substituting the lower limit 0 into the antiderivatives:
[4ln|4(0) + 1| - 4ln|1 + 0|] = [4ln|1| - 4ln|1|] = 0
Step 6: Calculate the final result
Now, subtract the lower limit value from the upper limit value:
[4ln|5| - 4ln|2|] - 0 = 4ln|5| - 4ln|2|
Therefore, the value of the integral ∫[0,1](12/(4x^2+5x+1))dx is 4ln|5| - 4ln|2|.
To evaluate the given integral, we can start by factoring the denominator of the integrand.
The denominator is 4x^2 + 5x + 1. To factorize it, we need to find two numbers whose product is 4 * 1 = 4 and whose sum is 5.
The numbers that satisfy this condition are 4 and 1. So we can write the denominator as (4x + 1)(x + 1).
Now we can rewrite the integral as follows:
∫(12/(4x^2 + 5x + 1))dx = ∫(12/((4x + 1)(x + 1)))dx
Next, we can use partial fractions to break down the integrand into simpler fractions. We want to decompose it in the form:
A/(4x + 1) + B/(x + 1)
To find the values of A and B, we need to make a common denominator on the right side and equate the numerators:
12 = A(x + 1) + B(4x + 1)
Expanding the right side:
12 = Ax + A + 4Bx + B
Rearranging the terms:
12 = (A + 4B)x + (A + B)
Since the left side is a constant and the right side has an x term, the coefficients of x on both sides must be equal. The constant terms on both sides must also be equal.
Equating the coefficients:
A + 4B = 0
A + B = 12
From the first equation, we get A = -4B. Substituting this into the second equation:
-4B + B = 12
Simplifying:
-3B = 12
Dividing both sides by -3:
B = -4
Substituting this value of B back into A = -4B:
A = -4(-4) = 16
Now we have the decomposition:
12/(4x^2 + 5x + 1) = 16/(4x + 1) - 4/(x + 1)
We can now rewrite the integral using these decomposed fractions:
∫(12/(4x^2 + 5x + 1))dx = ∫(16/(4x + 1) - 4/(x + 1))dx
Using the linearity property of integration, we can split this into two separate integrals:
= ∫(16/(4x + 1))dx - ∫(4/(x + 1))dx
Now we can evaluate each integral individually.
To evaluate ∫(16/(4x + 1))dx, we can use the substitution method. Let u = 4x + 1. Then du/dx = 4 and dx = du/4.
Substituting these in:
= ∫(16/u)(du/4) = 4∫du/u = 4ln|u| + C1
Since u = 4x + 1, we can substitute back:
= 4ln|4x + 1| + C1
To evaluate ∫(4/(x + 1))dx, we can use the natural logarithm identity ln|x| = ln(x) - ln|x|. In this case, ln|x + 1| = ln(x + 1) - ln|x + 1|.
Rewriting the integral:
= 4∫(1/(x + 1))dx = 4ln|x + 1| - 4ln|x + 1| + C2
Simplifying:
= 0 + 4ln|x + 1| + C2
Combining both results:
∫(12/(4x^2 + 5x + 1))dx = 4ln|4x + 1| + 4ln|x + 1| + C
where C is the constant of integration.
12/(4x^2+5x+1) = 12/(4x+1)(x+1) = 16/(4x+1) - 4/(x+1)
So, now it's easy.
4log(4x+1) - 4log(x+1) = 4log (4x+1)/(x+1)
now just evaluate at 0,1 to get
4log(5/2)-4log(1/1) = 4log(5/2)