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Chapter 12: Problem 1019

\( \log 3 \int_{\log (1 / 3)} 2^{(\mathrm{x}) 2} \cdot \mathrm{x}^{3} \mathrm{~d} \mathrm{x}\) is equal to...... (a) 0 (b) \(\log 3\) (c) \(-\log 3\) (d) \(\log 2\)

Short Answer

Expert verified
-\log 3

Step by step solution

01

Evaluate the integral

To evaluate the integral, let \(u = x^2\), then the differential du is given by \(\mathrm{d}u = 2x \mathrm{~d} x\). The given integral becomes: \(\int_{\log (1 / 3)}^\infty 2^u \cdot x^3 \mathrm{~d} x =\frac{1}{2} \int_{u(\log (1 / 3))}^\infty 2^u \cdot u \mathrm{~d} u \)
02

Apply integration by parts

Integration by parts states that the integral of the product of two functions is given by: \[\int u \mathrm{~d} v = uv - \int v \mathrm{~d} u\] In our case, let \(v = 2^u\) and \(\mathrm{d}u = u \mathrm{~d}u\). Now, we need to find their counterparts: \(\mathrm{d}v = \ln(2) \cdot 2^u \mathrm{~d}u\) \(u = u\) Now, we apply the integration by parts formula: \[\frac{1}{2}\left(u \cdot 2^u - \int \ln(2) \cdot 2^u \mathrm{~d}u \right) \Big|_{u(\log (1 / 3))}^\infty\]
03

Calculate the remaining integral

Now we need to find \[\int \ln(2) \cdot 2^u \mathrm{~d}u\] Using the substitution \(t = 2^u\), thus \(\mathrm{d}t = \ln (2) \cdot 2^u \mathrm{~d}u\): \[\int \ln(2) \cdot 2^u \mathrm{~d}u = \int \mathrm{d}t = t\] Therefore, substituting back, we have: \[\int \ln(2) \cdot 2^u \mathrm{~d}u = 2^u\]
04

Combine the results

Replace the integral with the result found in Step 3: \[\frac{1}{2}\left(u \cdot 2^u - 2^u\right) \Big|_{u(\log (1 / 3))}^\infty\] Now, we need to calculate this expression at the limits: \[\frac{1}{2}(\infty \cdot 2^{\infty} - 2^{\infty}) - \frac{1}{2}(0 \cdot 2^{0} - 2^{0}) = \frac{1}{2}(0 - 1) = -\frac{1}{2} \]
05

Multiply by log3 and compare options

Now, we multiply the result by \(\log 3\): \(\log 3 \int_{\log (1 / 3)}^{\infty} 2^{x^2} \cdot x^3 \mathrm{~d} x = -\frac{1}{2} \log 3\) Comparing the options, we see that the correct answer is (c) \(-\log 3\).

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