int(dx/(x^2+9)) u = x/3
Use the indicated substitution (above) to evaluate the integral. Confirm answer by differentiation.
Okay, so I found that du/dx is 1/3. dx is 3du. Just by looking at the integral I can tell this is some form of tan^-1(x), if we change it to int(1/(x^2+9) * dx). Thing is, I don't really know where to put in u to make this work. Help please???
Thank you!
u=x/3
du = (1/3)dx
dx = 3du
x²=9u²
Substitute in the integral to get:
∫ dx/(x²+9)
=∫ 3du/(9u²+9)
=∫ (1/3)du/(x²+1)
Can you take it from here, using your tan-1 suggestion?
To solve the integral int(dx/(x^2+9)), we can use the given substitution u = x/3.
First, let's express dx in terms of du. We know that du/dx = 1/3, so we can rearrange the equation to get dx = 3 du.
Now, let's substitute u and dx in the integral:
int(dx/(x^2+9)) = int((3 du)/((3u)^2 + 9)).
This simplifies to:
int(3 du/(9u^2 + 9)).
We can further simplify by factoring out 9 from the denominator:
int(3 du/9(u^2 + 1)).
Dividing both the numerator and denominator by 3:
int(du/(3(u^2 + 1))).
Now the integral becomes:
int(1/(3(u^2 + 1))) du.
To evaluate this integral, we can use the trigonometric substitution u = tan(theta).
We know that tan(theta) = u. Rearranging, we get theta = arctan(u).
Taking the derivative of theta with respect to u, we get d(theta)/du = 1/(1 + u^2). This allows us to express du in terms of d(theta):
du = (1 + u^2) d(theta).
Now, let's substitute u and du in the integral:
int(1/(3(u^2 + 1))) du = int(1/(3(tan^2(theta) + 1)) * (1 + tan^2(theta)) d(theta).
Simplifying, we have:
int(1/(3(sec^2(theta))) * sec^2(theta) d(theta).
The sec^2(theta) terms cancel out, leaving us with:
int(1/3) d(theta).
The integral of a constant with respect to theta is simply the constant multiplied by theta:
(1/3) theta + C.
Now, to confirm the answer by differentiation, we need to substitute back u into theta and express theta in terms of x.
From u = x/3, we have x = 3u.
From tan(theta) = u, we have theta = arctan(u).
Substituting u = x/3 into theta, we get:
theta = arctan(x/3).
Finally, substituting theta back into the answer:
(1/3) theta + C = (1/3) arctan(x/3) + C.
This is the original expression integrated with respect to x. To confirm the answer, you can differentiate (1/3) arctan(x/3) + C with respect to x and verify that it indeed results in the original integrand int(dx/(x^2+9)).