Evaluate the Integral integral of 19/(2+5y) with respect to y

Problem

(∫_^)(19/(2+5*y)*d(y))

Solution

1. Factor out the constant 19 from the integral to simplify the expression.

(∫_^)(19/(2+5*y)*d(y))=19*(∫_^)(1/(2+5*y)*d(y))

2. Identify the form of the integral, which matches the rule (∫_^)(1/(a*x+b)*d(x))=1/a*ln(a*x+b)+C

3. Apply the integration rule where a=5 and b=2

19*(∫_^)(1/(2+5*y)*d(y))=19⋅1/5*ln(2+5*y)+C

4. Simplify the resulting expression by multiplying the constants.

19/5*ln(2+5*y)+C

Final Answer

(∫_^)(19/(2+5*y)*d(y))=19/5*ln(2+5*y)+C

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