1/a arctan u/a +c
To evaluate the integral of 1/a arctan(u/a) with respect to u, we can use the substitution method
To evaluate the integral of 1/a arctan(u/a) with respect to u, we can use the substitution method. Let’s set u/a = v, which implies u = av and du = a dv.
Substituting these values back into the integral, we have:
∫(1/a) arctan(u/a) du = ∫(1/a) arctan(v) (a dv)
Simplifying, we have:
(1/a) * a * ∫arctan(v) dv
The ‘a’ in the numerator cancels out with the ‘a’ in the denominator, leading to:
∫arctan(v) dv
At this point, we can integrate the function arctan(v) with respect to v. The integral of arctan(v) is v * arctan(v) – ln|v + 1| + c, where c is the constant of integration.
Therefore, the final answer is:
∫(1/a) arctan(u/a) du = u * arctan(u/a) – ln|u/a + 1| + c
Note that ‘c’ represents the constant of integration, as there are infinite antiderivatives of any given function.
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