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Suppose the rate of growth of the fly population is given by $g(t)={e}^{0.01t},$ and the initial fly population is 100 flies. How many flies are in the population after 15 days?
There are 116 flies.
Evaluate the definite integral using substitution: ${\int}_{1}^{2}\frac{{e}^{1\text{/}x}}{{x}^{2}}dx}.$
This problem requires some rewriting to simplify applying the properties. First, rewrite the exponent on e as a power of x , then bring the x ^{2} in the denominator up to the numerator using a negative exponent. We have
Let $u={x}^{\mathrm{-1}},$ the exponent on e . Then
Bringing the negative sign outside the integral sign, the problem now reads
Next, change the limits of integration:
Notice that now the limits begin with the larger number, meaning we must multiply by −1 and interchange the limits. Thus,
Evaluate the definite integral using substitution: ${\int}_{1}^{2}\frac{1}{{x}^{3}}{e}^{4{x}^{\mathrm{-2}}}}dx.$
${\int}_{1}^{2}\frac{1}{{x}^{3}}{e}^{4{x}^{\mathrm{-2}}}}dx=\frac{1}{8}\left[{e}^{4}-e\right]$
Integrating functions of the form $f\left(x\right)={x}^{\mathrm{-1}}$ result in the absolute value of the natural log function, as shown in the following rule. Integral formulas for other logarithmic functions, such as $f(x)=\text{ln}\phantom{\rule{0.1em}{0ex}}x$ and $f(x)={\text{log}}_{a}x,$ are also included in the rule.
The following formulas can be used to evaluate integrals involving logarithmic functions.
Find the antiderivative of the function $\frac{3}{x-10}.$
First factor the 3 outside the integral symbol. Then use the u ^{−1} rule. Thus,
See [link] .
Find the antiderivative of $\frac{1}{x+2}.$
$\text{ln}\left|x+2\right|+C$
Find the antiderivative of $\frac{2{x}^{3}+3x}{{x}^{4}+3{x}^{2}}.$
This can be rewritten as $\int \left(2{x}^{3}+3x\right){\left({x}^{4}+3{x}^{2}\right)}^{\mathrm{-1}}dx}.$ Use substitution. Let $u={x}^{4}+3{x}^{2},$ then $du=4{x}^{3}+6x.$ Alter du by factoring out the 2. Thus,
Rewrite the integrand in u :
Then we have
Find the antiderivative of the log function ${\text{log}}_{2}x.$
Follow the format in the formula listed in the rule on integration formulas involving logarithmic functions. Based on this format, we have
Find the antiderivative of ${\text{log}}_{3}x.$
$\frac{x}{\text{ln}\phantom{\rule{0.1em}{0ex}}3}\left(\text{ln}\phantom{\rule{0.1em}{0ex}}x-1\right)+C$
[link] is a definite integral of a trigonometric function. With trigonometric functions, we often have to apply a trigonometric property or an identity before we can move forward. Finding the right form of the integrand is usually the key to a smooth integration.
Find the definite integral of ${\int}_{0}^{\pi \text{/}2}\frac{\text{sin}\phantom{\rule{0.1em}{0ex}}x}{1+\text{cos}\phantom{\rule{0.1em}{0ex}}x}dx.$
We need substitution to evaluate this problem. Let $u=1+\text{cos}\phantom{\rule{0.1em}{0ex}}x,,$ so $du=\text{\u2212}\text{sin}\phantom{\rule{0.1em}{0ex}}x\phantom{\rule{0.2em}{0ex}}dx.$ Rewrite the integral in terms of u , changing the limits of integration as well. Thus,
Then
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