How do you use u-substitution with 20sin(x^2/35)?
I will assume that you are finding the derivative with respect to x.
let u = x^2/35 or u = (1/35)x^2
let y = sin u
dy/du = cos u , and du/dx = (2/35)x
dy/dx = (dy/du)(du/dx)
= (cos u)(2/35)x
= (2x/35)cos (x^2/35)
your aim will be to reach that final line directly.
To use u-substitution with the function 20sin(x^2/35), we follow these steps:
Step 1: Choose a substitution
We set u equal to the expression inside the sine function, so u = x^2/35.
Step 2: Find du/dx.
To find du/dx, we differentiate both sides of the equation u = x^2/35 with respect to x. In this case, du/dx is simply (2x)/35.
Step 3: Solve for dx.
To solve for dx, we rearrange the previous equation: dx = (35/2x) du.
Step 4: Substitute the u value and dx in the original integral.
The original integral can now be expressed in terms of u: ∫20sin(x^2/35) dx = ∫20sin(u) (35/2x) du.
Step 5: Simplify the integral.
We can simplify the integral further by rearranging the terms and canceling out common factors:
∫20sin(u) (35/2x) du = 35/2 ∫(20/x)sin(u) du.
Step 6: Integrate the simplified integral.
With the simplified integral, we can now integrate the function with respect to u: 35/2 ∫(20/x)sin(u) du = 35/2 ∫(20/x) (-cos(u)) du.
Step 7: Substitute back in terms of x.
Since we substituted u = x^2/35 in the beginning, we need to substitute back in terms of x. By rearranging the initial substitution, we have x = √(35u).
Step 8: Evaluate the integral.
We can now evaluate the integral using the substitution x = √(35u):
35/2 ∫(20/x) (-cos(u)) du = -700/√35 ∫cos(u) du.
Step 9: Integrate and simplify.
Integrating -700/√35 ∫cos(u) du gives us -700/√35 sin(u) + C, where C is the constant of integration.
Finally, we substitute back u = x^2/35:
-700/√35 sin(x^2/35) + C.
So, by using u-substitution, the integral of 20sin(x^2/35) with respect to x is -700/√35 sin(x^2/35) + C.