Math  /  Calculus

Question11 a Find 2xdx\int \frac{2}{x} d x.

Studdy Solution

STEP 1

1. We are asked to find the indefinite integral of the function 2x\frac{2}{x}.
2. The variable of integration is xx.
3. The function 2x\frac{2}{x} is continuous for x0x \neq 0.

STEP 2

1. Identify the basic integral form that matches the given function.
2. Apply the integration rule for the identified form.
3. Simplify the result and include the constant of integration.

STEP 3

Identify the basic integral form:
The function 2x\frac{2}{x} can be rewritten as 21x2 \cdot \frac{1}{x}, which matches the standard integral form 1xdx\int \frac{1}{x} \, dx.

STEP 4

Apply the integration rule:
The integral of 1x\frac{1}{x} with respect to xx is lnx\ln |x|. Therefore, the integral of 21x2 \cdot \frac{1}{x} is:
2xdx=21xdx=2lnx\int \frac{2}{x} \, dx = 2 \int \frac{1}{x} \, dx = 2 \ln |x|

STEP 5

Simplify the result and include the constant of integration:
The indefinite integral is:
2xdx=2lnx+C\int \frac{2}{x} \, dx = 2 \ln |x| + C
where CC is the constant of integration.
The solution to the integral is:
2lnx+C\boxed{2 \ln |x| + C}

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