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Section Anti-differentiation and the Indefinite Integral

We might denote the derivative of the function \(f(x) = x^2\) as any of:

\begin{equation*} f', \; \; \frac{df}{dx}, \; \; (x^2)', \; \; \mbox{or} \; \; \frac{d}{dx} (x^2). \end{equation*}

If \(f\) is a function, then any function with derivative \(f\) is called an anti-derivative of \(f.\) For the anti-derivative of \(f\) we may write any of:

\begin{equation*} \int f, \; \; \int f(x) \; dx, \; \; \mbox{or} \; \; \int x^2 \; dx. \end{equation*}

Any function which has one anti-derivative has infinitely many anti-derivatives. To indicate all anti-derivatives of \(f(x) = x^2\) we will write:

\begin{equation*} \int f(x) \; dx = \int x^2 \; dx = \frac{1}{3} x^3 + c, \ \ \ c \in \re. \end{equation*}

We would read this notation as “the indefinite integral (or anti-derivative) of \(x^2\) with respect to \(x\) is \(\frac{1}{3}x^3+c\) where \(c\) represents an arbitrary constant.” It is misleading that mathematicians speak of “the indefinite integral” when in fact it is not a single function, but a class of functions. You may interpret \(\int\) and “dx” as representing the beginning and end of the function we wish to find the anti-derivative of. We call this process integrating the function. The function to be integrated is called the integrand.

Problem 4.17.

Evaluate the indicated indefinite integrals and then state the Sum Rule for Indefinite Integrals based on your results.

  1. \(\dsp \int 3x^4 + 5\; dx\)

  2. \(\dsp \int 3x^4 \; dx + \int 5 \; dx\)

Problem 4.18.

Evaluate the indicated indefinite integrals and then state the Constant Multiple Rule for Indefinite Integrals based on your results.

  1. \(\dsp{ \int 5 \sin(x) \; dx }\)

  2. \(\dsp{ 5 \int \sin(x) \; dx }\)

Problem 4.19.

Evaluate the following indefinite integrals.

  1. \(\dsp \int (s^3+s)^3\; ds\)

  2. \(\dsp \int t(t^{1/2}+8)\; dt\)

  3. \(\dsp \int \frac{x+x^3}{x^5} \; dx\)

  4. \(\dsp \int 3\cos(x)-2\sin(4x) + e^{4x} \; dx\)

Like differentiation, anti-differentiation becomes more challenging when the Chain Rule appears in the problem.

Example 4.20.

Evaluate \(\dsp \int x^2(x^3+10)^9\; dx.\)

Our first guess will be \(g(x) = (x^3 + 10)^{10}.\) Then \(g'(x) = 10(x^3+10)^9 \cdot 3x^2 = 30x^2(x^3+10)^9.\) Our answer is correct except for the 30 in front. We take care of this by modifying our first guess to \(g(x) = \frac{1}{30}(x^3 + 10)^{10}.\) Now \(g'(x) = \frac{1}{30}\cdot 10(x^3 + 10)^{10} \cdot 3x^2 = x^2(x^3+10)^9\) and so our second guess is one anti-derivative of the integrand. Adding “+ c” to the end yields all possible anti-derivatives.

Problem 4.21.

Evaluate the following indefinite integrals.

  1. \(\dsp \int(x+2)(3x^2 + 12x - 5)^{121} \; dx\)

  2. \(\dsp \int(x+2)\sqrt[4]{3x^2+12x-5} \; dx\)

  3. \(\dsp \int p^2 \sqrt[3]{p^3+1} \; dp\)

  4. \(\dsp \int \frac{3\cos(\sqrt{t})}{\sqrt{t}} \; dt\)

  5. \(\dsp \int \frac{x}{(3x^2+4)^5} \; dx\)

  6. \(\dsp \int \frac{x}{3x^2+4} \; dx\)

Problem 4.22.

These look quite similar. Try both. What's the difference?

  1. \(\dsp \int \frac{\sin(t)}{\cos^2(t)} \; dt\)

  2. \(\dsp \int \frac{\sin(t)}{\cos(t^2)} \; dt\)

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