∫1/(x2 − 2x + 8)3/2 dx
Find the following integral ? How , is it by substitution !!!!!!!!
I do not know but suspect you mean
∫1/(x^2 − 2x + 8)^(3/2) dx
∫(x2 − 2x + 8)^-1.5 dx
http://www.wolframalpha.com/widget/widgetPopup.jsp?p=v&id=7c9206fea2b96c1f22626b03b18327b3&title=Math+Help+Boards:+Indefinite+Integral+Calculator&theme=blue
says:
(x-1)/[7 sqrt(x^2-2x+8) ]
Note that you have
∫1/(x^2 − 2x + 8)^(3/2) dx
= ∫1/((x-1)^2 + 7)^(3/2) dx
If x-1 = √7 tanu
(x-1)^2 + 7 = 7 sec^2 u
dx = √7 sec^2 u du
and you have
∫1/(√7 secu)^3 * √7 sec^2 u du
= ∫ 1/7 cosu du
= 1/7 sinu
= 1/(7 csc u)
= 1/(7 √(1+cot^2 u))
= 1/(7 √(1 + (7/(x-1)^2))
= 1/(7/(x-1) √((x-1)^2 + 7)
= 1/ 7√(x^2-2x+8)
Simpler than I'd have expected.
To find the integral ∫1/(x^2 - 2x + 8)^(3/2) dx, we can use the technique of integration by substitution. Here are the steps:
Step 1: Complete the Square
First, we need to complete the square in the denominator to simplify the expression. The given denominator is x^2 - 2x + 8.
To complete the square, we want to rewrite the expression as a perfect square. We can do this by adding and subtracting a constant term. In this case, we need to add 1 to both sides of the equation:
x^2 - 2x + 8 = (x^2 - 2x + 1) + 7
= (x - 1)^2 + 7
Now, our integral becomes ∫1/((x - 1)^2 + 7)^(3/2) dx.
Step 2: Substitute a New Variable
Next, we make a substitution to simplify the integral. Let u = x - 1. This implies du = dx, as the derivative of u with respect to x is 1.
Using the substitution, we can rewrite the integral as follows:
∫1/((x - 1)^2 + 7)^(3/2) dx = ∫1/(u^2 + 7)^(3/2) du.
Step 3: Evaluate the Integral
Now, the integral is in the form ∫1/(u^2 + a^2)^(3/2) du, which can be solved using a standard integral formula:
∫1/(u^2 + a^2)^(3/2) du = 1/a^2 * ∫u/√(u^2 + a^2) du.
In our case, a^2 = 7, so we have:
∫1/(u^2 + 7)^(3/2) du = 1/7 * ∫u/√(u^2 + 7) du.
Step 4: Solve the Integral
Now, we can solve the integral 1/7 * ∫u/√(u^2 + 7) du.
To solve this integral, we use another substitution. Let v = u^2 + 7. This implies dv = 2u du, or du = dv/(2u).
Using the new substitution, we can rewrite the integral as follows:
1/7 * ∫u/√(u^2 + 7) du = 1/7 * ∫(1/2v) dv.
Simplifying further, we have:
1/7 * ∫(1/2v) dv = 1/14 * ∫(1/v) dv = 1/14 * ln|v| + C,
where ln|v| is the natural logarithm of the absolute value of v, and C is the constant of integration.
Step 5: Substitute Back and Simplify
We now substitute back into the original variables. Recall that v = u^2 + 7, and u = x - 1.
Substituting back, we get:
1/14 * ln|v| + C = 1/14 * ln|u^2 + 7| + C.
Finally, we replace u with x - 1:
∫1/(x^2 - 2x + 8)^(3/2) dx = 1/14 * ln|(x - 1)^2 + 7| + C,
where C is the constant of integration.
That's it! We have now found the integral of 1/(x^2 - 2x + 8)^(3/2) dx.