I need $u$ , I $du$ .

Definite Integrals and $u$ -Substitution

7-72.

Use any of the techniques you have learned so far to evaluate this integral.
Explain and show how you know your answer is correct.

$\int ( \frac { 3 } { 2 } x ^ { 4 } - 3 x ) ^ { 5 } ( 2 x ^ { 3 } - 1 ) d x$

7-73.

Use the substitution method to evaluate the following integrals. State the substitution, $u$ , used to evaluate the integral.

$\int 6 \operatorname { sin } ( 5 x ^ { 3 } ) \cdot x ^ { 2 } d x$

$\int e ^ { ( 5 x ^ { 10 } - 2 ) } \cdot x ^ { 9 } d x$

$\int x ^ { 3 } ( 7 x ^ { 4 } - 3 ) ^ { 5 } d x$

$\int \operatorname { sin } ( x ^ { 4 } - 3 ) \operatorname { cos } ^ { 5 } ( x ^ { 4 } - 3 ) x ^ { 3 } d x$

$\int \frac { x ^ { 2 } } { x ^ { 3 } - 1 } d x$

$\int \frac { 1 } { x ^ { 2 } } \sqrt { \frac { 1 } { x } + 1 } d x$

7-74.

Remember Greta? She is now trying to calculate the area under $f(x) = x^3(6x^4 - 3)^5$ over the interval $–1 ≤ x ≤ 0$ . She set up an integral, used $u$ -substitution, but she did not show all of her steps.

Check Greta’s work. Show any missing steps. If you find an error, correct it and explain.
Problem:
$\int _ { - 1 } ^ { 0 } x ^ { 3 } ( 6 x ^ { 4 } - 3 ) ^ { 5 } d x$
Step 1:
$\int _ { - 1 } ^ { 0 } \frac { 1 } { 24 } u ^ { 5 } d u$
Step 2:
$\frac { 1 } { 24 } ( \frac { 1 } { 6 } u ^ { 6 } | _ { - 1 } ^ { 0 } ) = - \frac { 1 } { 144 }$
Now, without using $u$ -substitution, use your calculator to verify Greta’s answer:
$\int _ { - 1 } ^ { 0 } x ^ { 3 } ( 6 x ^ { 4 } - 3 ) ^ { 5 } d x =$
Greta must have made a mistake! Her teammate Hansol noticed something interesting about Step 1 of Greta’s work: the new integrand is an odd function, $\frac { 1 } { 24 }u^5$ . Without showing any steps, Hansol promptly rewrote the integral with new bounds. $\int _ { 3 } ^ { - 3 } \frac { 1 } { 24 } u ^ { 5 } d u$ . Evaluate this definite integral. Does this answer agree with your answer in part (b)?
In part (c), Hansol changed the bounds to $–3$ and $3$ , but there are other bounds that will result in an area of $0 \text{ un}^2$ . Show the steps that Hansol used to obtain his bounds.

7-75.

In the following problems, rewrite each integral in terms of $u$ . Be careful of the bounds. You do not need to evaluate the integrals.

$\int _ { 1 } ^ { 3 } 2 x ( x ^ { 2 } + 1 ) ^ { 4 } d x$

$\int _ { 0 } ^ { \pi / 4 } \operatorname { sec } ^ { 2 } ( x ) ( \operatorname { tan } ( x ) + 3 ) d x$

7-76.

BAD GIRL!

Ms. D’s famous dog, Basil, is very naughty! One day, Basil grabs the edge of the tablecloth and backs away at $15$ cm/sec. This causes a glass on the table near the other end of the tablecloth to move to the edge and fall off and shatter. Basil’s mouth is $12$ cm above the floor, the table is $72$ cm high, and there is $250$ cm of tablecloth between Basil’s mouth and the glass. At the instant the glass reaches the table’s edge, how fast is it traveling?

7-77.

Differentiate each of the following functions. Homework Help ✎

$y = \tan^{-1}(2x)$

$y = \tan(2x)$

$y = \cot(2x)$

7-78.

Integrate. Homework Help ✎

$\int - \frac { 2 } { x } d x$

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

$\int _ { 0 } ^ { \pi } \pi ^ { x } d x$

7-79.

Grain pouring from a conveyor belt falls into a pile in the shape of a cone. The grain is falling at a rate of $8 \ \text{ m}^3\text{/min}$ and forms a pile such that the radius is always half the height. How fast is the height of the pile increasing when the radius is $6$ meters? Homework Help ✎

7-80.

THE WEDDING CAKE, Part Two

Kiki is still trying to decide on her wedding cake. She has decided to keep the same shape and size but instead have $8$ layers. Therefore, the diameters of the layers will be $16, 14, 12, 10, 8, 6, 4$ , and $2$ inches. Each layer for this cake will be $2$ inches tall. Homework Help ✎

Set up a Riemann sum to evaluate the volume of the cake.
Calculate the volume of this latest design.

7-81.

Yong Li’s initial deposit of $$1$ has grown to $$5$ in just $2$ years of continuously compounded interest! What is the annual interest rate? Homework Help ✎

7-82.

Remember our city that straddles the Newton River? The city is applying for National Parks and Recreation funds and needs to know the area of the land (shaded in the diagram) within the city limits.

Given the equations of the riverbanks, determine the area of the land. Homework Help ✎

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

7-83.

The limit below represents a definite integral. Fill in the blanks: Homework Help ✎

$\lim\limits _ { n \rightarrow \infty } \frac { 6 } { n } [ ( \frac { 1 } { n } + 3 ) +$ $( \frac { 2 } { n } + 3 ) + \ldots +$ $( \frac { n } { n } + 3 ) ] = \int _ { \boxed{\text{_}} } ^ { \boxed{\text{_}} } \boxed{\text{___}} d x$