Evaluate the Integral

Problem

(∫_^)((x*e(2*x))/((2*x+1)2)*d(x))

Solution

1. Rewrite the numerator to create a term that resembles the denominator by adding and subtracting 1/2

(∫_^)((1/2*(2*x+1−1)*e(2*x))/((2*x+1)2)*d(x))

2. Distribute the terms to split the integral into two separate parts.

(∫_^)((1/2*(2*x+1)*e(2*x))/((2*x+1)2)*d(x))−(∫_^)((1/2*e(2*x))/((2*x+1)2)*d(x))

3. Simplify the first integral by canceling the common factor (2*x+1)

(∫_^)((e(2*x))/(2*(2*x+1))*d(x))−(∫_^)((e(2*x))/(2*(2*x+1)2)*d(x))

4. Apply integration by parts to the first integral, (∫_^)(u*d(v)) where u=(e(2*x))/2 and d(v)=1/(2*x+1)*d(x)

d(u)=e(2*x)*d(x)

v=1/2*ln(2*x+1)

Note: This standard approach leads to a non-elementary integral. Instead, let u=1/(2*(2*x+1)) and d(v)=e(2*x)*d(x)

d(u)=−1/((2*x+1)2)*d(x)

v=1/2*e(2*x)

5. Substitute these into the integration by parts formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) for the first term.

(e(2*x))/(4*(2*x+1))−(∫_^)(1/2*e(2*x)*(−1/((2*x+1)2))*d(x))

6. Combine the result with the second integral from step 3.

(e(2*x))/(4*(2*x+1))+(∫_^)((e(2*x))/(2*(2*x+1)2)*d(x))−(∫_^)((e(2*x))/(2*(2*x+1)2)*d(x))

7. Cancel the matching integral terms and add the constant of integration.

(e(2*x))/(4*(2*x+1))+C

Final Answer

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

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