∫3 at top and 2 at bottom
x/x^2+1 dx = ?
if you could show steps that would be great thanks
actually theres an integral sign. at the top of the sign theres a 3. At the bottom of it, theres a 2.
You can type it this way:
∫(x/(x^2 + 1)) dx for x= 2 to 3
which is
[ (1/2) ln (x^2 + 1) ] from 2 to 3
= (1/2) ( ln 10 - ln 5)
or
(ln10 - ln5)/2
or
ln (10/5) / 2
= (1/2) ln 2
To solve the integral ∫(3 to 2) x / (x^2 + 1) dx, we can use a technique called partial fractions. Here are the steps to solve it:
Step 1: Express the fraction as partial fractions.
Start by factoring the denominator:
x^2 + 1 = (x + i)(x - i), where i is the imaginary unit (√(-1)).
Now, we can write the fraction as:
x / (x^2 + 1) = A / (x + i) + B / (x - i)
where A and B are unknown constants.
Step 2: Find the unknown constants.
To find A and B, we need to clear the denominators in the above equation. Multiply both sides of the equation by (x + i)(x - i):
x = A(x - i) + B(x + i)
Substitute x = i into the equation to solve for A:
i = A(i - i) + B(i + i)
i = 2Bi
Divide both sides by i:
1 = 2B
So, B = 1/2.
Substitute x = -i into the equation to solve for A:
-i = A(-i - i) + B(-i + i)
-i = -2Ai
Divide both sides by -i:
1 = 2A
So, A = 1/2.
Now, we have:
x / (x^2 + 1) = 1/2 / (x + i) + 1/2 / (x - i)
Step 3: Integrate each term separately.
∫(3 to 2) x / (x^2 + 1) dx = ∫(3 to 2) (1/2) / (x + i) dx + ∫(3 to 2) (1/2) / (x - i) dx
For the first integral, substitute u = x + i, and therefore dx = du:
∫(3 to 2) (1/2) / (x + i) dx = (1/2) ∫(3+ i to 2+ i) du / u
= (1/2) ln|u| ∣(3+ i to 2+ i)
= (1/2) ln|2+ i| - (1/2) ln|3+ i|
For the second integral, substitute u = x - i, and therefore dx = du:
∫(3 to 2) (1/2) / (x - i) dx = (1/2) ∫(3- i to 2- i) du / u
= (1/2) ln|u| ∣(3- i to 2- i)
= (1/2) ln|2- i| - (1/2) ln|3- i|
Step 4: Evaluate the definite integrals.
∫(3 to 2) x / (x^2 + 1) dx = (1/2) ln|2+ i| - (1/2) ln|3+ i| - (1/2) ln|2- i| + (1/2) ln|3- i|
Now, substitute the values of i = √(-1):
∫(3 to 2) x / (x^2 + 1) dx = (1/2) ln|2+ √(-1)| - (1/2) ln|3+ √(-1)| - (1/2) ln|2- √(-1)| + (1/2) ln|3- √(-1)|
Since √(-1) is equal to i, the above expression simplifies to:
∫(3 to 2) x / (x^2 + 1) dx = (1/2) ln|2+ i| - (1/2) ln|3+ i| - (1/2) ln|2- i| + (1/2) ln|3- i|
And this is the final solution using the partial fractions method.