How do I integrate this?
constant(1- z/[sqrt (z^2 + constant^2)]
i know i can take the constant out so I have
constant * integral 1- z/[sqrt (z^2 + constant^2)]
and I know the integral of
z/[sqrt (z^2 + constant^2)] is
[ln z^2 + constant^2]/2
I just need to know in general how to separate the 1 + , i.e.
integral of 1 + f(x) = ?
sorry i meant 1- f(x)
1-, not 1+
is it just integral 1 + integral f(x)
and the integral 1 = x?
Yes, the integral of 1(*dx) is just x.
To separate the integral of 1 + f(x), you can split it into two separate integrals and then evaluate each integral individually.
Let's say you have:
∫ (1 + f(x)) dx
You can rewrite it as:
∫ 1 dx + ∫ f(x) dx
The integral of 1 with respect to x is simply x + C, where C is the constant of integration. So, the first integral becomes:
∫ 1 dx = x + C1
For the second integral, you can use the given result for integrating f(x). Let's use F(x) to represent the antiderivative of f(x):
∫ f(x) dx = F(x) + C2
Now, you can combine the results of the two integrals to get:
∫ (1 + f(x)) dx = (x + C1) + (F(x) + C2)
Simplifying further, you can write it as:
∫ (1 + f(x)) dx = x + F(x) + C
Here, C is the constant of integration, which is the sum of C1 and C2, and F(x) represents the antiderivative of f(x).
So, in general, the integral of 1 + f(x) is equal to x plus the antiderivative of f(x), plus the constant of integration.