Evaluate the Integral integral of 1/(1+4x^2) with respect to x

Problem

(∫_^)(1/(1+4*x2)*d(x))

Solution

1. Identify the integral form as a variation of the standard arctangent integral (∫_^)(1/(1+u2)*d(u))=arctan(u)+C

2. Rewrite the denominator to express the term 4*x2 as a perfect square (2*x)2

(∫_^)(1/(1+(2*x)2)*d(x))

3. Apply a substitution by letting u=2*x which implies d(u)=2*d(x) or d(x)=1/2*d(u)

(∫_^)(1/(1+u2)⋅1/2*d(u))

4. Factor out the constant 1/2 from the integral.

1/2*(∫_^)(1/(1+u2)*d(u))

5. Integrate using the arctangent rule.

1/2*arctan(u)+C

6. Substitute back u=2*x to get the final expression in terms of x

1/2*arctan(2*x)+C

Final Answer

(∫_^)(1/(1+4*x2)*d(x))=1/2*arctan(2*x)+C

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