Evaluate the Integral

Problem

(∫_^)(1/(9*x2+16)*d(x))

Solution

1. Identify the form of the integral as a standard inverse tangent integral, which follows the pattern (∫_^)(1/(u2+a2)*d(u))=1/a*arctan(u/a)+C

2. Rewrite the denominator to identify u and a by expressing the terms as squares.

(∫_^)(1/((3*x)2+4)*d(x))

3. Substitute u=3*x which implies d(u)=3*d(x) or d(x)=1/3*d(u)

(∫_^)(1/(u2+4)⋅1/3*d(u))

4. Factor out the constant 1/3 from the integral.

1/3*(∫_^)(1/(u2+4)*d(u))

5. Apply the formula for the arctangent integral where a=4

1/3⋅1/4*arctan(u/4)+C

6. Simplify the coefficients and substitute 3*x back in for u

1/12*arctan((3*x)/4)+C

Final Answer

(∫_^)(1/(9*x2+16)*d(x))=1/12*arctan((3*x)/4)+C

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