a function $F$ such that $F' (x) = f (x)$ for all $x$ in the domain of $f$ is an antiderivative of $f$

the most general antiderivative of $f (x)$ is the indefinite integral of $f;$ we use the notation $\int f (x) d x$ to denote the indefinite integral of $f$

Calculus Volume 1 · 4.10 Antiderivatives

Antiderivatives: practice problems

31 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

Exercises 465
For the following exercises, show that $F (x)$ are antiderivatives of $f (x) .$
$F (x) = 5 x^{3} + 2 x^{2} + 3 x + 1, f (x) = 15 x^{2} + 4 x + 3$
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Exercises 467
For the following exercises, show that $F (x)$ are antiderivatives of $f (x) .$
$F (x) = x^{2} e^{x}, f (x) = e^{x} (x^{2} + 2 x)$
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Exercises 469
For the following exercises, show that $F (x)$ are antiderivatives of $f (x) .$
$F (x) = e^{x}, f (x) = e^{x}$
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Exercises 471
For the following exercises, find the general form for the antiderivative of the function.
$f (x) = e^{x} - 3 x^{2} + sin x$
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Exercises 473
For the following exercises, find the general form for the antiderivative of the function.
$f (x) = x - 1 + 4 sin (2 x)$
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Exercises 475
For the following exercises, find the general form for the antiderivative $F (x)$ of each function $f (x) .$
$f (x) = x + 12 x^{2}$
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Exercises 477
For the following exercises, find the general form for the antiderivative $F (x)$ of each function $f (x) .$
$f (x) = (x)^{3}$
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Exercises 479
For the following exercises, find the general form for the antiderivative $F (x)$ of each function $f (x) .$
$f (x) = \frac{x ^{1 / 3}}{x ^{2 / 3}}$
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Exercises 481
For the following exercises, find the general form for the antiderivative $F (x)$ of each function $f (x) .$
$f (x) = sec^{2} (x) + 1$
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Exercises 483
For the following exercises, find the general form for the antiderivative $F (x)$ of each function $f (x) .$
$f (x) = sin^{2} (x) cos (x)$
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Exercises 485
For the following exercises, find the general form for the antiderivative $F (x)$ of each function $f (x) .$
$f (x) = \frac{1}{2} csc^{2} (x) + \frac{1}{x ^{2}}$
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Exercises 487
For the following exercises, find the general form for the antiderivative $F (x)$ of each function $f (x) .$
$f (x) = 4 csc x cot x - sec x tan x$
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Exercises 489
For the following exercises, find the general form for the antiderivative $F (x)$ of each function $f (x) .$
$f (x) = \frac{1}{2} e^{- 4 x} + sin x$
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Exercises 491
For the following exercises, evaluate the integral.
$\int sin x d x$
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Exercises 493
For the following exercises, evaluate the integral.
$\int \frac{3 x ^{2} + 2}{x ^{2}} d x$
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Exercises 495
For the following exercises, evaluate the integral.
$\int (4 x + 4 x) d x$
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Exercises 497
For the following exercises, evaluate the integral.
$\int \frac{14 x ^{3} + 2 x + 1}{x ^{3}} d x$
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Exercises 499
For the following exercises, solve the initial value problem.
$f^{'} (x) = x^{- 3}, f (1) = 1$
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Exercises 501
For the following exercises, solve the initial value problem.
$f^{'} (x) = cos x + sec^{2} (x), f (\frac{π}{4}) = 2 + \frac{2}{2}$
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Exercises 503
For the following exercises, solve the initial value problem.
$f^{'} (x) = \frac{2}{x ^{2}} - \frac{x ^{2}}{2}, f (1) = 0$
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Exercises 505
For the following exercises, find two possible functions $f$ given the second- or third-order derivatives.
$f^{''} (x) = e^{- x}$
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Exercises 507
For the following exercises, find two possible functions $f$ given the second- or third-order derivatives.
$f ‴ (x) = cos x$
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Exercises 509
For the following exercises, find two possible functions $f$ given the second- or third-order derivatives.
A car is being driven at a rate of $40$ mph when the brakes are applied. The car decelerates at a constant rate of $10$ ft/sec². How long before the car stops?
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Exercises 511
For the following exercises, find two possible functions $f$ given the second- or third-order derivatives.
You are merging onto the freeway, accelerating from rest at a constant rate of $12$ ft/sec². How long does it take you to reach merging speed at $60$ mph?
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Exercises 513
For the following exercises, find two possible functions $f$ given the second- or third-order derivatives.
A car company wants to ensure its newest model can stop in $8$ sec when traveling at $75$ mph. If we assume constant deceleration, find the value of deceleration that accomplishes this.
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Exercises 515
For the following exercises, find the antiderivative of the function, assuming $F (0) = 0.$
[T] $f (x) = x^{2} + 2$
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Exercises 517
For the following exercises, find the antiderivative of the function, assuming $F (0) = 0.$
[T] $f (x) = sin x + 2 x$
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Exercises 519
For the following exercises, find the antiderivative of the function, assuming $F (0) = 0.$
[T] $f (x) = \frac{1}{( x + 1 ) ^{2}}$
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Exercises 521
For the following exercises, determine whether the statement is true or false. Either prove it is true or find a counterexample if it is false.
If $f (x)$ is the antiderivative of $v (x),$ then $2 f (x)$ is the antiderivative of $2 v (x) .$
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Exercises 523
For the following exercises, determine whether the statement is true or false. Either prove it is true or find a counterexample if it is false.
If $f (x)$ is the antiderivative of $v (x),$ then $f (x) + 1$ is the antiderivative of $v (x) + 1.$
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Review Exercises 545
Find the antiderivatives $F (x)$ of the following functions.
$g (x) = x - \frac{1}{x ^{2}}$
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Key terms · 3

antiderivative: a function $F$ such that $F' (x) = f (x)$ for all $x$ in the domain of $f$ is an antiderivative of $f$
indefinite integral: the most general antiderivative of $f (x)$ is the indefinite integral of $f;$ we use the notation $\int f (x) d x$ to denote the indefinite integral of $f$
initial value problem: a problem that requires finding a function $y$ that satisfies the differential equation $\frac{d y}{d x} = f (x)$ together with the initial condition $y (x_0) = y_0$

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Antiderivatives: practice problems

31 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

Exercises 465

For the following exercises, show that $F (x)$ are antiderivatives of $f (x) .$

$F (x) = 5 x^{3} + 2 x^{2} + 3 x + 1, f (x) = 15 x^{2} + 4 x + 3$

Exercises 467

For the following exercises, show that $F (x)$ are antiderivatives of $f (x) .$

$F (x) = x^{2} e^{x}, f (x) = e^{x} (x^{2} + 2 x)$

Exercises 469

For the following exercises, show that $F (x)$ are antiderivatives of $f (x) .$

$F (x) = e^{x}, f (x) = e^{x}$

Exercises 471

For the following exercises, find the general form for the antiderivative of the function.

$f (x) = e^{x} - 3 x^{2} + sin x$

Exercises 473

For the following exercises, find the general form for the antiderivative of the function.

$f (x) = x - 1 + 4 sin (2 x)$

Exercises 475