Try It Yourself#
Exercise 1#
Integral of a constant
Compute \(\displaystyle \int 2e ~dy\). Verify your answer by computing its derivative.
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Answer: \(2ey + C\)
Exercise 2#
Integral of a sum of power functions
Compute \(\displaystyle \int 3 x^{5/4} + \frac{7}{\sqrt{x}}+ \frac{1}{6x^5} ~dx\). Verify your answer by computing its derivative.
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Answer: \(\dfrac{4}{3}x^{9/4} + 14\sqrt{x} - \dfrac{1}{24x^4} + C\)
Exercise 3#
Integral of a product
Compute \(\displaystyle \int (e^{3x} + 4)(e^{-5x} - 20) ~dx\). Verify your answer by computing its derivative.
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Answer: \(-\dfrac{1}{2}e^{-2x} - \dfrac{20}{3}e^{3x} - \dfrac{4}{5}e^{-5x} - 80x + C\)
Exercise 4#
Integral of a polynomial divided by a power function
Compute \(\displaystyle \int \frac{10x^3 - 2x^2 + 3x-25 }{x} ~dx\). Verify your answer by computing its derivative.
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Answer: \(\dfrac{10}{3}x^3 - x^2 + 3x - 25\ln|x| + C\)
Exercise 5#
Initial value problem
Find \(f(x)\) such that \( f'(x) = e^{7x} + 2x^3 +3 \) and \(f(0)=2\). Verify your answer by computing its derivative.
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Answer: \(\dfrac{1}{7}e^{7x} + \dfrac{1}{2}x^4 + 3x + \dfrac{13}{7}\)
Exercise 6#
Total cost
The marginal cost function associated with producing \(x\) croissants is given by
where \(C'(x)\) is measured in dollars/unit and \(x\) denotes the number of croissants.
If the daily fixed costs incurred in the production is \(\$400\), find the total cost \(C(x)\) incurred in producing the first 100 units of the day.
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Answer: $3900
Exercise 7#
Integration by substitution
Compute \(\displaystyle \int \frac{5x-3}{(5x^2-6x+15)^4} ~dx\). Verify your answer by computing its derivative.
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Answer: \(-\dfrac{1}{6(5x^2 - 6x + 15)^3} + C\)
Exercise 8#
Integration by substitution
Compute \(\displaystyle \int \frac{x}{x-2} ~dx\). Verify your answer by computing its derivative.
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Answer: \(x + 2\ln|x-2| + C\)
Exercise 9#
Integration by substitution
Compute \(\displaystyle \int \frac{e^{3x}}{e^{3x}-4} ~dx\). Verify your answer by computing its derivative.
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Answer: \(\dfrac{1}{3}\ln|e^{3x}-4| + C\)
Exercise 10#
Initial value problem
Find \(f(x)\) such that \(f'(x) = (5x+1)(5x^2+2x-1)^2\) and \(f(1) = 15\). Verify your answer by computing its derivative.
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Answer: \(\dfrac{1}{6}(5x^2+2x-1)^3 - 21\)
Exercise 11#
Integration by substitution
Compute \(\displaystyle \int 3x ~\sqrt[]{x^2+10} ~dx\). Verify your answer by computing its derivative.
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Answer: \((x^2+10)^{3/2} + C\)
Exercise 12#
Integration by substitution
Compute \(\displaystyle \int 15x^3 ~\sqrt[]{x^2+10} ~dx\). Verify your answer by computing its derivative.
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Answer: \(3(x^2+10)^{5/2} - 50(x^2+10)^{3/2} + C\)