Integrating a Complicated Function by Parts
To solve the integral provided in your image, we can separate it into two integrals and solve each one individually. Here is the integral:
∫(√(2x) - 3 + 7x^(3/4) + (2 / (3x + 1))) dx
Now we can separate it into individual integrals:
∫√(2x) dx - ∫3 dx + ∫7x^(3/4) dx + ∫(2 / (3x + 1)) dx
Let's integrate each term one by one.
1. For ∫√(2x) dx, let u = 2x. Then, du = 2 dx, or (1/2)du = dx.
The integral becomes (1/2)∫u^(1/2) du, which evaluates to (1/2) * (2/3)u^(3/2) + C = (1/3) * (2x)^(3/2) + C1.
2. For ∫3 dx, the integral is simply 3x + C2.
3. For ∫7x^(3/4) dx, the integral is 7 * (4/7)x^(7/4) + C = 4x^(7/4) + C3.
4. For ∫(2 / (3x + 1)) dx, let u = 3x + 1. Then, du = 3 dx, or (1/3)du = dx.
The integral becomes (2/3)∫(1/u) du, which evaluates to (2/3)ln|u| + C = (2/3)ln|3x + 1| + C4.
Now combine all the results back together to get the final solution:
(1/3) * (2x)^(3/2) + 3x + 4x^(7/4) + (2/3)ln|3x + 1| + C
Where C is the constant of integration that is the sum of all individual constants C1, C2, C3, and C4.
