FIND EACH INDEFINITE INTERGRAL BY THE SUBSTITUTION METHOD
DX/1+5X
To find the indefinite integral of dx/(1+5x) using the substitution method, follow these steps:
Step 1: Choose a substitution
Let u = 1 + 5x. This means that du/dx = 5, and solving for dx, we get dx = du/5.
Step 2: Substitute the variables
Replace dx with du/5 and replace 1 + 5x with u in the integral.
The integral becomes: (1/5) ∫ du/u
Step 3: Evaluate the integral
Integrating 1/u with respect to u gives us ln|u|, where ln denotes the natural logarithm.
Therefore, the integral becomes: (1/5) ln|u| + C
Step 4: Undo the substitution
Remember that u = 1 + 5x. Substitute u back into the expression:
(1/5) ln|1+5x| + C
And that's the indefinite integral of dx/(1+5x) using the substitution method.