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Chapter 16: Problem 49

Evaluate \(2 \int_{0}^{1} e^{x} \sin x d x\) a) \(1.82\) b) \(1.94\) c) \(2.05\) d) \(2.16\)

Short Answer

Expert verified
Answer: The value of the integral is approximately (b) \(1.94\).

Step by step solution

01

Choose Functions u and dv

Choose the functions \(u\) and \(dv\) as follows: \(u = e^x\) (since its derivative will simplify to itself) \(dv = \sin x dx\) (since we know how to integrate the sin function)
02

Find Derivatives and Integrals for u and dv

Differentiate \(u\) with respect to \(x\) and integrate \(dv\) with respect to \(x\) \(du = e^x dx\) \(v = -\cos x\)
03

Apply Integration by Parts

Apply the integration by parts formula to the problem \(\int_{0}^{1} e^x \sin x dx = [e^x (-\cos x)]_{0}^{1} - \int_{0}^{1} -e^x \cos x dx\)
04

Calculate Initial Result

Evaluate the expression: \([e^x (-\cos x)]_{0}^{1} = (e^1 (-\cos 1)) - (e^0 (-\cos 0))\) Now we need to calculate the remaining integral \(\int_{0}^{1} -e^x \cos x dx\) by applying integration by parts again.
05

Apply Integration by Parts the Second Time

Choose the functions \(u\) and \(dv\) as follows: \(u = -e^x\) \(dv = \cos x dx\) Differentiate \(u\) with respect to \(x\) and integrate \(dv\) with respect to \(x\): \(du = -e^x dx\) \(v = \sin x\) Apply the integration by parts formula again: \(\int_{0}^{1} -e^x \cos x dx = [-e^x (\sin x)]_{0}^{1} - \int_{0}^{1} \sin x (-e^x) dx\)
06

Calculate Remaining Terms

Evaluate the expression: \([-e^x (\sin x)]_{0}^{1} = (-e^1 (\sin 1)) - (-e^0 (\sin 0))\) Now, we need to find the remaining integral targeted in previous steps.
07

Combine Results

Substitute the calculated values back into the original integral: \(\int_{0}^{1} e^x \sin x dx = (e^1 (-\cos 1)) - (e^0 (-\cos 0)) - (-e^1 (\sin 1)) + (e^0 (\sin 0)) - \int_{0}^{1} \sin x (-e^x) dx\) Notice that the expression now involves only the same integral expression that we started with: \(\int_{0}^{1} e^x \sin x dx = e - 1 + \sin 1 - \int_{0}^{1} e^x \sin x dx\)
08

Solve for the Integral

Rearrange to solve for the integral. \(\int_{0}^{1} e^x \sin x dx = \frac{e - 1 + \sin 1}{2}\) Now, multiply the result by 2 as given in the original problem: \(2\int_{0}^{1} e^x \sin x dx = e - 1 + \sin 1\)
09

Compare to Options

Compare the final result to the given options and choose the correct one. a) \(1.82\) b) \(1.94\) c) \(2.05\) d) \(2.16\) Using a calculator, we get: \(e - 1 + \sin 1 ≈ 1.94\) Therefore, the correct answer is (b) \(1.94\).

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