integrate 2/((2x-7)^2) from x=4 to 6
Let 2x -7 = u
so dx = du/2
Integrand becomes
2/u^2*(du/2) = du/u^2 from u = 1 to 5.
= -1/u @ u = 5 -
(-1/u @ u = 1)
= 1 - 1/5 = 4/5
2/((2x-7)^2)
= 2(2x-7)^-2
integral of that is
-(2x-7)^-1 or -1/(2x-7)
and its value form x=4 to 6
= -1/5 + 1
= 4/5
To find the definite integral ∫[4 to 6] [2/((2x-7)^2)] dx, you can follow these steps:
Step 1: Rewrite the integral
The given integral is ∫[4 to 6] [2/((2x-7)^2)] dx.
Step 2: Simplify the integrand
Start by simplifying the expression (2x-7)^2.
(2x-7)^2 = (2x-7) * (2x-7) = 4x^2 - 28x + 49
Now rewrite the integral as: ∫[4 to 6] [2/(4x^2 - 28x + 49)] dx.
Step 3: Apply the linear substitution
To evaluate the integral, we can use a linear substitution. Let u = 2x - 7, and solve for dx in terms of du:
du/dx = 2
dx = du/2
Now substitute u and dx in the integral:
∫[4 to 6] [2/(4x^2 - 28x + 49)] dx = ∫[4 to 6] [2/(4(u+7) - 28(u+7) + 49)] (du/2)
Step 4: Simplify the integral
Simplify the expression inside the integral by factoring out a common factor from the denominator:
= ∫[4 to 6] [2/(4u - 28 + 49)] (du/2)
= ∫[4 to 6] [2/(4u + 21)] (du/2)
= (1/2) ∫[4 to 6] [1/(2u + 10.5)] du
Step 5: Evaluate the integral
Now, the integral becomes a simple logarithmic integral. Apply the formula:
∫[4 to 6] [1/(2u + 10.5)] du = (1/2) ln|2u + 10.5| evaluated from 4 to 6
Substitute the limits of integration into the formula:
(1/2) ln|2(6) + 10.5| - (1/2) ln|2(4) + 10.5|
Step 6: Calculate the result
Now, you can calculate the result by substituting the values into the expression:
(1/2) ln|12 + 10.5| - (1/2) ln|8 + 10.5|
(1/2) ln|22.5| - (1/2) ln|18.5|
Use the properties of logarithms to simplify further:
(1/2) ln(22.5) - (1/2) ln(18.5)
Finally, use a calculator to calculate the result approximately.
(1/2) ln(22.5) ≈ 0.80472
(1/2) ln(18.5) ≈ 0.65809
The final result is approximately 0.80472 - 0.65809 ≈ 0.14663.