Evaluate the Integral integral of 4/(x^2+9) with respect to x

Problem

(∫_^)(4/(x2+9)*d(x))

Solution

1. Factor out the constant 4 from the integral to simplify the expression.

(∫_^)(4/(x2+9)*d(x))=4*(∫_^)(1/(x2+9)*d(x))

2. Identify the standard integral form (∫_^)(1/(x2+a2)*d(x))=1/a*arctan(x/a)+C where a2=9 which means a=3

a=3

3. Apply the formula for the arctangent integral using a=3

4*(∫_^)(1/(x2+3)*d(x))=4*(1/3*arctan(x/3))+C

4. Simplify the expression by multiplying the constants.

4/3*arctan(x/3)+C

Final Answer

(∫_^)(4/(x2+9)*d(x))=4/3*arctan(x/3)+C

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