Question

\(∫_0^1 (5xe^{2x}-tan(π/4)) dx=\)

• A. \( (\dfrac{5}{4}) + e^2\)
• B. \( (\dfrac{5}{4}) + e^2 +\dfrac{9}{4}\)
• C. \( (\dfrac{3}{4}) + e^2 +\dfrac{1}{4}\)
• D. \( (\dfrac{1}{4}) + e^2 +\dfrac{5}{4}\)
• E. \( (\dfrac{5}{4}) + e^2 +\dfrac{1}{4}\)

Answer 1

Answer: (E)

Step 1: Split the integral into two parts: \[ \int_0^1 \left(5xe^{2x} - \tan\frac{\pi}{4}\right) dx = \int_0^1 5xe^{2x} dx - \int_0^1 \tan\frac{\pi}{4} dx \]

Step 2: Evaluate the second integral (constant term): \[ \tan\frac{\pi}{4} = 1 \] \[ \int_0^1 1 dx = 1 \]

Step 3: Evaluate the first integral using integration by parts: Let \( u = 5x \), \( dv = e^{2x}dx \) Then \( du = 5dx \), \( v = \frac{1}{2}e^{2x} \) \[ \int 5xe^{2x} dx = \frac{5x}{2}e^{2x} - \int \frac{5}{2}e^{2x} dx = \frac{5x}{2}e^{2x} - \frac{5}{4}e^{2x} + C \]

Step 4: Evaluate from 0 to 1: \[ \left. \left( \frac{5x}{2}e^{2x} - \frac{5}{4}e^{2x} \right) \right|_0^1 = \left( \frac{5}{2}e^2 - \frac{5}{4}e^2 \right) - \left( 0 - \frac{5}{4} \right) = \frac{5}{4}e^2 + \frac{5}{4} \]

Step 5: Combine results: \[ \left( \frac{5}{4}e^2 + \frac{5}{4} \right) - 1 = \frac{5}{4}e^2 + \frac{1}{4} \]

Conclusion: The correct answer is \(\left( \frac{5}{4}e^2 + \frac{1}{4} \right)\).

Answer 2

Let the given integral be

\[ I = \int_0^1 \left( 5xe^{2x} - \tan\left(\frac{\pi}{4}\right) \right) dx \]

Since \( \tan\left(\frac{\pi}{4}\right) = 1 \), we have

\[ I = \int_0^1 (5xe^{2x} - 1) dx = 5 \int_0^1 xe^{2x} dx - \int_0^1 1 dx \]

We use integration by parts to evaluate \( \int_0^1 xe^{2x} dx \). Let \( u = x \) and \( dv = e^{2x} dx \). Then \( du = dx \) and \( v = \frac{1}{2}e^{2x} \).

\[ \int_0^1 xe^{2x} dx = \left[ \frac{1}{2}xe^{2x} \right]_0^1 - \int_0^1 \frac{1}{2}e^{2x} dx = \frac{1}{2}e^2 - \frac{1}{2} \left[ \frac{1}{2}e^{2x} \right]_0^1 = \frac{1}{2}e^2 - \frac{1}{4}(e^2 - 1) = \frac{1}{4}e^2 + \frac{1}{4} \]

Therefore,

\[ I = 5 \left( \frac{1}{4}e^2 + \frac{1}{4} \right) - \int_0^1 1 dx = \frac{5}{4}e^2 + \frac{5}{4} - [x]_0^1 = \frac{5}{4}e^2 + \frac{5}{4} - 1 = \frac{5}{4}e^2 + \frac{1}{4} \]

Thus,

\[ I = \frac{5}{4}e^2 + \frac{1}{4} \]

Final Answer: The final answer is \( {A} \).