how to i evaluate the integral of
dx/ (x^2 +9) with a u substitution of x/3?
Can someone show this step by step?
To evaluate the integral ∫ dx / (x^2 + 9) using the u-substitution method, follow these step-by-step instructions:
Step 1: Choose a substitution
Let u = x/3 be the substitution chosen. This substitution will help simplify the integral.
Step 2: Calculate du/dx
Differentiate u = x/3 with respect to x to find du/dx:
du/dx = 1/3
Step 3: Solve for dx
Rearrange the equation du/dx = 1/3 to solve for dx:
dx = 3 du
Step 4: Substitute the variables
Now substitute the variables in the integral:
∫ dx / (x^2 + 9) = ∫ (3 du) / (9u^2 + 9)
= ∫ du / (3u^2 + 3)
Step 5: Simplify the integral
Simplify the integrand:
∫ du / (3u^2 + 3) = (1/3) ∫ du / (u^2 + 1)
Step 6: Evaluate the integral
To evaluate the integral ∫ du / (u^2 + 1), this integral is a known form - the arctangent function. The integral becomes:
(1/3) ∫ du / (u^2 + 1) = (1/3) arctan(u) + C
Step 7: Substitute back u = x/3
Finally, substitute back the original substitution to recover the original variable, x:
(1/3) arctan(u) + C = (1/3) arctan(x/3) + C
Therefore, the solution to the integral of dx / (x^2 + 9) using the u-substitution of x/3 is (1/3) arctan(x/3) + C, where C is the constant of integration.