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Question

Evaluate the integral ∫2𝑑𝑥/(5𝑥+10),

by making the appropriate substitution: u = 5x+10Correct  

∫2𝑑𝑥/(5𝑥+10) =    + C

Answer 1

1. Let u = 5x + 10.
2. Differentiate u with respect to x: du/dx = 5, so dx = du / 5.
3. Substitute into the integral: ∫ 2dx / (5x + 10) = ∫ (2 * du / 5) / u.
4. Factor out constants: (2/5) ∫ du / u.
5. The integral of (1/u) is ln|u|, so: (2/5) ln|u| + C.
6. Substitute u = 5x + 10 back in: (2/5) ln|5x + 10| + C.

Final answer: (2/5) ln|5x + 10| + C.

Answer 2

2 ∫ dx / (5x+10}

Let u = 5x+10. Then du = 5dx. So, dx = (1/5)du

So, we have (2/5) ∫ du / u = (2/5) ln l u l + C = (2/5) ln l 5x + 10 l + C