integrate (3+x)x^(1/2)dx=?
please help I have no idea how to integrate this problem
just expand the expression
(3+x)x^(1/2) = 3x^(1/2) + x^(3/2)
then just use the power rule on each term.
but with integration there is no power rule wouldn't I have to use substitution?
nope.
Just as
d/dx x^n = n x^(n-1)
∫n x^(n-1) dx = x^n
or, as it is ually written,
∫x^n dx = 1/(n+1) x^(n+1)
So,
∫x^(1/2) dx = 2/3 x^(3/2)
see when you take the derivative of that, that you get x^(1/2)
To integrate the expression (3+x)x^(1/2)dx, you can use the technique of u-substitution.
Let's break down the steps to solve this problem:
1. Identify the u and du terms: Choose a part of the expression to substitute as "u." In this case, let's pick u = x^(1/2) and du = (1/2)x^(-1/2)dx.
2. Rewrite the expression using u and du: To replace x^(1/2)dx, substitute u for x^(1/2) and du for (1/2)x^(-1/2)dx. This gives us ∫(3+u)du.
3. Simplify the expression: Distribute the integral sign and simplify the expression. This gives us ∫3du + ∫udu.
4. Integrate the expression: Now we can integrate each term separately. The integral of 3du is 3u, and the integral of udu is (1/2)u^2.
5. Combine the results: After integrating each term, we have 3u + (1/2)u^2 + C, where C is the constant of integration.
Therefore, the final result is ∫(3+x)x^(1/2)dx = 3(x^(1/2)) + (1/2)x(x^(1/2)) + C.
Hope this explanation helps!