Evaluate the Integral

Problem

(∫_^)(5/(9*x2+49)*d(x))

Solution

1. Factor out the constant from the numerator to simplify the integrand.

(∫_^)(5/(9*x2+49)*d(x))=5*(∫_^)(1/(9*x2+49)*d(x))

2. Rewrite the denominator in the form u2+a2 to prepare for the arctangent integration formula.

5*(∫_^)(1/((3*x)2+7)*d(x))

3. Identify the substitution u=3*x which implies d(u)=3*d(x) or d(x)=1/3*d(u)

5*(∫_^)(1/(u2+7)⋅1/3*d(u))

4. Apply the standard integration formula (∫_^)(1/(u2+a2)*d(u))=1/a*arctan(u/a)+C

5/3⋅1/7*arctan(u/7)+C

5. Substitute u=3*x back into the expression and multiply the constants.

5/21*arctan((3*x)/7)+C

Final Answer

(∫_^)(5/(9*x2+49)*d(x))=5/21*arctan((3*x)/7)+C

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