Question
Integration (solve step by step)
(1/x+2/x^3+2)dx
(1/x+2/x^3+2)dx
Answers
GPT 3.5
To integrate (1/x + 2/x^3 + 2)dx, we can break it down into three separate integrals:
1. Integral of 1/x dx:
∫(1/x)dx = ln|x| + C
2. Integral of 2/x^3 dx:
∫(2/x^3)dx = ∫2x^(-3)dx
= 2∫x^(-3) dx
= 2 * (-1/2)x^(-2) + C
= -x^(-2) + C
= -1/x^2 + C
3. Integral of 2 dx:
∫2dx = 2x + C
Putting it all together:
∫(1/x + 2/x^3 + 2)dx = ln|x| - 1/x^2 + 2x + C
Therefore, the integral of (1/x + 2/x^3 + 2)dx is ln|x| - 1/x^2 + 2x + C.
1. Integral of 1/x dx:
∫(1/x)dx = ln|x| + C
2. Integral of 2/x^3 dx:
∫(2/x^3)dx = ∫2x^(-3)dx
= 2∫x^(-3) dx
= 2 * (-1/2)x^(-2) + C
= -x^(-2) + C
= -1/x^2 + C
3. Integral of 2 dx:
∫2dx = 2x + C
Putting it all together:
∫(1/x + 2/x^3 + 2)dx = ln|x| - 1/x^2 + 2x + C
Therefore, the integral of (1/x + 2/x^3 + 2)dx is ln|x| - 1/x^2 + 2x + C.