What have $u$ done for me lately?

Integration With $u$ -Substitution

7-62.

(NOT SO) HARD-LOOKING INTEGRALS, Part Two

Here are some integrals that are similar to those you worked on previously. Again, they have a similar structure. As you work, focus on this question: How do you decide which constant(s) you will need to multiply by?

$\int 6 x ^ { 2 } ( x ^ { 3 } + 5 ) d x$

$\int \frac { 3 } { 4 } x ^ { 3 } ( x ^ { 4 } + 1 ) ^ { - 1 / 2 } d x$

7-63.

Antiderivatives (like your answers to problem 7-62) can get messy when the Chain Rule is involved, only in reverse. At times, it is hard to keep everything straight. You need to keep track of which is the inside function, which is the outside function, and what constants you need to multiply or divide by. What a mess!

One technique that helps organize this work is called substitution. The steps for the substitution method are outlined in the following Math Notes box. Use the steps to evaluate the following integrals.

$\int \sin(x^5) · x^4dx$

$\int (−3x^4 + 2)^5 · x^3dx$

$\int \sec(10x^3) \tan(10x^3) · x^2dx$

$\int x^5(−3 + 5x^6)^9dx$

$\int 25x^4 · 5^{10x^5+1}dx$

$\int (x^3 + 1)(x^4 + 4x + 5)^{8/5}dx$

Review the problems you just integrated using $u$ -substitution. Are there any expressions where more than one possible expression for $u$ could have been defined? Choose at least two problems to integrate again using a different $u$ . Did you get the same answer?

Math Notes

Substitution Method for Integration

Can you tell just by looking at the integral at right that it is a “reverse Chain Rule” integral? How?

Assume the inside function is $\frac { 3 } { 2 }x^4 − 3x$ . Call the inside function $\boldsymbol u$ .

The goal is to replace all the $x$ ’s with $u$ ’s:

In order to replace $dx$ , differentiate the $\boldsymbol u$ equation and
solve for $dx$ .

Now substitute the expressions for $u$ and $dx$ into the integral:

If $u$ was chosen correctly, then the integral will be written completely in terms of $u$ . No $x$ ’s will remain.

Now the general antiderivative can be found, in terms of $u$ .

Finally, if you choose, you can substitute $\frac { 3 } { 2 }x^4 − 3x$ and write the general antiderivative in terms of $x$ .

$\large\left. \begin{array} { l } { \int ( \frac { 3 } { 2 } x ^ { 4 } - 3 x ) ^ { 5 } ( 2 x ^ { 3 } - 1 ) d x } \\ { \quad \text { Let } u = \frac { 3 } { 2 } x ^ { 4 } - 3 x } \\ { \text { Then } \frac { d u } { d x } = 6 x ^ { 3 } - 3 } \\ { \qquad \left. \begin{array}{l}{ d u = ( 6 x ^ { 3 } - 3 ) d x }\\{ d x = \frac { d u } { ( 6 x ^ { 3 } - 3 ) } }\end{array} \right. } \end{array} \right.$

$\large\left. \begin{array} { c } { \int u ^ { 5 } ( 2 x ^ { 3 } - 1 ) \frac { d u } { ( 6 x ^ { 3 } - 3 ) } } \\ { \int u ^ { 5 } \cdot \frac { ( 2 x ^ { 3 } - 1 ) } { ( 6 x ^ { 3 } - 1 ) } d u } \\ { \int u ^ { 5 } \frac { ( 2 x ^ { 3 } - 1 ) } { 3 ( 2 x ^ { 3 } - 1 ) } d u } \\ { \int u ^ { 5 } \cdot \frac { 1 } { 3 } d u } \\ { \frac { 1 } { 18 } u ^ { 6 } + C } \\ { \frac { 1 } { 18 } ( \frac { 3 } { 2 } x ^ { 4 } - 3 x ) ^ { 6 } + C } \end{array} \right.$

7-64.

Right triangle, vertical leg labeled, x, vertical leg extended above vertex opposite horizontal leg, to point labeled, window, distance from bottom right angle vertex to window labeled, 300 feet, horizontal leg labeled, 800 feet, angle opposite vertical leg labeled, theta. As a fire truck is parked outside a building, its rotating red beacon makes a red dot as it shines on the building wall.

If the beacon rotates at $\frac { \pi } { 10 }$ rad/sec, how fast is the red dot moving along the wall when it hits a window $300$ feet away from the wall’s closest point? Use the information in diagram at right to answer this question.
Are you surprised by how fast the red dot is moving? Why is the speed so high?

7-65.

Once again, Greta is in a pickle! She is trying to solve the following equation and knows she can use $u$ -substitution. Homework Help ✎

$3 \tan^2(x) - \tan(x) - 10 = 0$

Greta thinks if she lets $u = \tan(x)$ she can write the equation as a quadratic equation. Show Greta she is correct.
Use your answer to part (a) to solve for $x$ .

7-66.

Remember Eric and the $16$ -foot tall lamppost? If Eric (who is $5$ feet tall) walks away from the lamppost at a rate of $4$ ft/sec, at what rate is the tip of his shadow moving away from the lamppost? Draw a diagram before you start this problem. Homework Help ✎

7-67.

The city planning commission has set aside funds to build a bridge going north and south across the Newton River. To save money, they have agreed to build the shortest bridge possible. A map for the $8$ -mile wide city is shown at right. The equations below represent the riverbanks where x is the number of miles east of City Hall and y is the number of miles north of City Hall. Determine the location for the shortest bridge that will span the Newton River.

North Bank: $y = \cos( \frac { \pi x } { 2 } )+6$ South Bank: $y = \ln(9 - x)$ Homework Help ✎

7-68.

No calculator! Determine where the following functions are not continuous and/or non-differentiable. Homework Help ✎

Compute without a calculator

$f(x) =\left\{ \begin{array} { l l } { \frac { 1 } { 2 } x ^ { 2 } + \frac { 3 } { 2 } } & { \text { for } x \leq - 1 } \\ { | 3 - x | } & { \text { for } x > - 1 } \end{array} \right.$
$g(m) =\left\{ \begin{array} { l l l } { \frac { 2 } { m - 3 } } & { \text { for } } & { m < 2 } \\ { - \sqrt { 6 - m } } & { \text { for } } & { 2 \leq m < 5 } \\ { \frac { 1 } { 2 } m - \frac { 7 } { 2 } } & { \text { for } } & { m \geq 5 } \end{array} \right.$

7-69.

While chasing a rabbit, a greyhound starts from rest and accelerates at $6$ ft/sec² until it reaches its maximum speed. Homework Help ✎

How long does it take for the greyhound to reach its maximum speed of $30$ ft/sec?
How far did the greyhound have to travel to catch the rabbit if it took a total of $8$ seconds to catch it?

7-70.

After analyzing a function $f$ , Edwin knew that $f^\prime(3) = 0$ and $f^{\prime\prime}(3) = -2$ . He therefore concluded that $f$ had an absolute maximum at $x = 3$ . Edwina, his girlfriend, is not so sure. Explain why Edwina is unwilling to accept Edwin’s conclusion. Homework Help ✎

7-71.

$F(x) =\int _ { 0 } ^ { x } f ( t ) d t$ for the function $y = f(t)$ graphed at right. Homework Help ✎

When is $F$ at a maximum on $[0, 15]$ ?
Is $F$ increasing, decreasing, or both over the interval $(0, 8)$ ?
Evaluate $F(4)$ , $F(10)$ , and $F(15)$ .
List the interval(s) on $[0, 15]$ for which $F^{\prime\prime}(x) > 0$ .

First quadrant continuous piecewise, labeled f of t, left curve, decreasing concave down curve, labeled circular, starting @ (0, comma 4), ending @ (4, comma 0), center segment from (4, comma 0), to (8, comma 6), right segment from (8, comma 6) to (15, comma 0).

Calculus Third Edition

Substitution Method for Integration