Try It Yourself#
Exercise 1#
Integral of a product of a power function and an eponential function
Compute \(\displaystyle \int xe^{x/3} ~dx\).
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Answer: \(3(x-3)e^{x/3} + C\)
Exercise 2#
Integral of a product of a power function and a logarithmic function
Compute \(\displaystyle \int \sqrt{x}\ln(x) ~dx\).
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Answer: \(\dfrac{2}{3} x^{3/2} \ln(x) - \dfrac{4}{9}x^{3/2} + C\)
Exercise 3#
Integral of a product of a polynomial and an exponential function
Compute \(\displaystyle \int (x^2+2x)e^x ~dx\). (Hint: Use integration by parts twice.)
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Answer: \(x^2e^x+C\)
Exercise 4#
Definite integral using integration by parts
Evaluate \(\displaystyle \int_1^e (\ln x)^2 ~dx\). (Hint: Use integration by parts with \(u = (\ln x)^2\) and \(dv = dx\).)
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Answer: \(e-2\)
Exercise 5#
Present and future value of an income stream
Suppose an investment is expected to generate income at the rate of \(R(t) = 10 + 7t\) thousands of dollars per year for the next 20 years. Find the present and future values from this investment if the prevailing interest rate is 5% per year compounded continuously.
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Answer: Present value: $866,299.24, Future value: $2,354,845.49
Exercise 6#
Improper integral
Evaluate \(\displaystyle \int_3^{\infty} e^{-7x} ~dx\), if it exists.
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Answer: \(\dfrac{1}{7e^{21}}\)
Exercise 7#
Improper integral
Evaluate \(\displaystyle \int_4^{\infty} \frac{e^{-\sqrt{x}}}{\sqrt{x}} ~dx\), if it exists. (Hint: Use a substitution to compute \(\displaystyle \int \frac{e^{-\sqrt{x}}}{\sqrt{x}} ~dx\).)
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Answer: \(2/e^2\)
Exercise 8#
Improper integral
Evaluate \(\displaystyle \int_2^{\infty} \frac{1}{(2x-3)^{1.5}} ~dx\), if it exists.
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Answer: \(1\)
Exercise 9#
Improper integral
Evaluate \(\displaystyle \int_6^{\infty} \frac{1}{(6x-35)^{0.5}} ~dx \), if it exists.
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Answer: Does not exist
Exercise 10#
Present value of a perpetuity
In order to create an endowment which pays $100 per week, in perpetuity, how much money must be invested today if the annual interest rate is 5% compounded continuously?
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Answer: $104,000