Evaluate the following indefinite integral by using the given substitution to reduce the integral to standard form
integral cos(9x) dx, u=9x
To evaluate the given integral, we can use the substitution method. Given that u = 9x, we can rewrite the integral as follows:
∫cos(9x) dx
Now, differentiate both sides of the equation u = 9x with respect to x to find du/dx:
du/dx = 9
To continue, we need to find dx in terms of du. Rearranging the equation above, we have:
dx = du/9
Now substitute the values of cos(9x) and dx into the integral:
∫cos(9x) dx = ∫cos(u) * (du/9)
Next, we can simplify the integral further:
(1/9) ∫cos(u) du
Since the integral of cos(u) is sin(u) plus a constant, the final solution is:
(1/9) * sin(u) + C
However, we need to express the solution in terms of x. To do this, we substitute back u = 9x:
(1/9) * sin(9x) + C
Therefore, the value of the given integral is:
∫cos(9x) dx = (1/9) * sin(9x) + C