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Mathematics LibreTexts

4.0E Exercises

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    Exercise \(\PageIndex{1}\)

    1. Define the term “antiderivative” in your own words

    2. Is it more accurate to refer to “the” antiderivative of \(f(x)\) or “an” antiderivative of \(f(x)\)?

    3. Use your own words to define the indefinite integral of \(f(x)\).

    4. Fill in the blanks: “Inverse operations do the ____ things in the _____ order.”

    5. What is an “initial value problem”?

    6. The derivative of a position function is a _____ function.

    7. The antiderivative of an acceleration function is a ______ function.


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    Exercise \(\PageIndex{2}\)

    Evaluate the indefinite integrals:

    1. \(\int 3x^3 \,dx\)

    2. \(\int x^8 \,dx\)

    3. \(\int (10x^2-2) \,dx\)

    4. \(\int \,dt\)

    5. \(\int 1 \,ds\)

    6. \(\int \frac{1}{3t^2}\, dt\)

    7. \(\int \frac{1}{t^2}\, dt\)

    8. \(\int \frac{1}{\sqrt{x}}\, dx\)

    9. \(\int \sec^2 \theta\, d\theta\)

    10. \(\int \sin \theta\, d\theta\)

    11. \(\int (\sec x \tan x +\csc x \cot x )\, dx\)

    12. \(\int 5e^\theta\, d\theta\)

    13. \(\int 3^t\, dt\)

    14. \(\int \frac{5^t}{2}\, dt\)

    15. \(\int (2t+3)^2\, dt\)

    16. \(\int (t^2+3)(t^3-2t)\, dt\)

    17. \(\int x^2x^3\, dx\)

    18. \(\int e^\pi\, dx\)

    19. \(\int a\, dx\)


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    Exercise \(\PageIndex{3}\)

    This problem investigates why Theorem 35 states that \(\int \frac{1}{x}\,dx = \ln |x|+C\).
    (a) What is the domain of \(y=\ln x\)?
    (b) Find \(\frac{d}{dx}(\ln x)\).
    (c) What is the domain of \(y=\ln (-x)\)?
    (d) Find \(\frac{d}{dx}\left ( (\ln (-x)\right )\).
    (e) You should find that \(1/x\) has two types of antiderivatives, depending on whether \(x>0\) or \(x<0\). In one expression, give a formula for \(\int \frac{1}{x}\,dx\) that takes these different domains into account, and explain your answer.


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    Exercise \(\PageIndex{4}\)

    Find \(f(x)\) described by the given initial value problem.

    1. \(f'(x)=\sin x\text{ and }f(0)=2\)

    2. \(f'(x)=5e^x\text{ and }f(0)=10\)

    3. \(f'(x)=4x^3-3x^2\text{ and }f(-1)=9\)

    4. \(f'(x)=\sec^2 x\text{ and }f(\pi/4)=5\)

    5. \(f'(x)=7^x\text{ and }f(2)=1\)

    6. \(f''(x)=5\text{ and }f'(0)=7,f(0)=3\)

    7. \(f''(x)=7x\text{ and }f'(1)=-1,f(1)=10\)

    8. \(f''(x)=5e^x\text{ and }f'(0)=3,f(0)=5\)

    9. \(f''(\theta)=\sin \theta \text{ and }f'(\pi)=2,f(\pi)=4\)

    10. \(f''(x)=24x^2+2^x-\cos x \text{ and }f'(0)=5,f(0)=0\)

    11. \(f''(x)=0\text{ and }f'(1)=3,f(1)=1\)


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    Exercise \(\PageIndex{5}\)

    Use information gained from the first and second derivative to sketch \(f(x)=\frac{1}{e^x+1}\).


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    Exercise \(\PageIndex{6}\)

    Given \(y=x^2e^x\cos x\), find \(dy\).


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