Chapter 17: Problem 1

Find (a) \(\int x \sin (2 x) \mathrm{d} x\), (b) \(\int t \mathrm{e}^{3 t} \mathrm{~d} t\) (c) \(\int x \cos x \mathrm{~d} x\).

Short Answer

Expert verified

Question: Evaluate the following integrals using integration by parts: a) \(\int x \sin(2x) \, dx\) b) \(\int t e^{3t} \, dt\) c) \(\int x \cos x \, dx\) Answer: a) \(-\frac{1}{2} x \cos(2x) -\frac{1}{4}\sin(2x) + C\) b) \(\frac{1}{3}t e^{3t} - \frac{1}{9}e^{3t} + C\) c) \(x\sin x + \cos x + C\)

Step by step solution

a) Integrate \(\int x \sin(2x) \, dx\)

To solve the integral \(\int x \sin(2x) \, dx\), we can use integration by parts: Choose \(u=x\) and \(dv = \sin(2x) \, dx\). Now, we need to find \(du\) and \(v\): \(du = dx, \; v = \int \sin(2x) \, dx = -\frac{1}{2}\cos(2x) + C\). Now apply the integration by parts formula: \(\int x \sin(2x) \, dx = uv - \int v \, du = -\frac{1}{2} x \cos(2x) - \int (-\frac{1}{2}\cos(2x)) \, dx\) To finish the integral, we need to integrate \(-\frac{1}{2}\cos(2x)\): \(-\frac{1}{4}\sin(2x) + C'\) Combining the results, we obtain: \(\int x \sin(2x) \, dx = -\frac{1}{2} x \cos(2x) -\frac{1}{4}\sin(2x) + C\)

b) Integrate \(\int t e^{3t} \, dt\)

To solve the integral \(\int t e^{3t} \, dt\), we can use integration by parts: Choose \(u=t\) and \(dv=e^{3t}\,dt\). Now, we need to find \(du\) and \(v\): \(du=dt, \; v=\int e^{3t} \, dt=\frac{1}{3}e^{3t}+C\). Now apply the integration by parts formula: \(\int t e^{3t} \, dt = uv - \int v \, du = \frac{1}{3}t e^{3t} - \int \left(\frac{1}{3}e^{3t}\right)\, dt\) To finish the integral, we need to integrate \(\frac{1}{3}e^{3t}\): \(\frac{1}{9}e^{3t}+C'\) Combining the results, we obtain: \(\int t e^{3t} \, dt = \frac{1}{3}t e^{3t} - \frac{1}{9}e^{3t} + C\)

c) Integrate \(\int x \cos x \, dx\)

To solve the integral \(\int x \cos x \, dx\), we can use integration by parts: Choose \(u=x\) and \(dv=\cos x \, dx\). Now, we need to find \(du\) and \(v\): \(du=dx, \; v=\int \cos x \, dx=\sin x+C\). Now apply the integration by parts formula: \(\int x \cos x \, dx = uv - \int v \, du = x\sin x - \int \sin x \, dx\) To finish the integral, we need to integrate \(\sin x\): \(-\cos x + C'\) Combining the results, we obtain: \(\int x \cos x \, dx = x\sin x + \cos x + C\)

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Find (a) \(\int x \sin (2 x) \mathrm{d} x\), (b) \(\int t \mathrm{e}^{3 t} \mathrm{~d} t\) (c) \(\int x \cos x \mathrm{~d} x\).

Short Answer

Step by step solution

a) Integrate \(\int x \sin(2x) \, dx\)

b) Integrate \(\int t e^{3t} \, dt\)

c) Integrate \(\int x \cos x \, dx\)

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