What else can $u$ $du$ ?

Varied Integration Techniques

7-84.

Use $u$ -substitution to rewrite the following definite integrals entirely in terms of $u$ . Be sure to change your bounds when necessary. Verify your work by evaluating both forms of the integral on your calculator.

$\int _ { 0 } ^ { - 1 } 3 x ^ { 2 } \sqrt { x ^ { 3 } + 1 } d x$

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

$\int _ { 0 } ^ { \pi / 4 } \operatorname { sin } ^ { 2 } ( 2 x ) \operatorname { cos } ( 2 x ) d x$

7-85.

$U$ -substitution can be used to “undo” the Chain Rule, and, it can also be used in other situations as well. Develop a method to evaluate the following integrals. Hint: Let $u$ equal the expression in the denominator.

$\int \frac { x + 5 } { x - 3 } d x$

$\int _ { - 1 } ^ { 2 } \frac { x + 5 } { x - 3 } d x$

Demonstrate another method (not $u$ -substitution) that you can use to evaluate the integrals above.

7-86.

Practice your integration skills by evaluating these integrals. Use $u$ -substitution when convenient.

$\int \frac { - 10 } { ( 3 x - 5 ) ^ { - 3 } } d x$

$\int \frac { \operatorname { sin } ( x ) } { \operatorname { cos } ( x ) } d x$

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

$\int \frac { - 10 } { 3 x - 5 } d x$

$\int \tan(x)dx$

$\int \frac { \operatorname { cos } ( 5 x ^ { 3 } - 9 x ^ { 5 } ) } { \operatorname { sin } ^ { 2 } ( 5 x ^ { 3 } - 9 x ^ { 5 } ) } ( x ^ { 2 } - 3 x ^ { 4 } ) d x$

7-87.

Use $u$ -substitution to solve the differential equation $\frac{dy}{dx}=\cot(x)$ for $y$ .

7-88.

Gary pumps gas into his gas tank at a constant rate of $6$ ounces per second. He does not realize it, but a hole in his tank allows gas to leak out at a rate of $\sqrt { t + 1 }$ ounces per second.

How many ounces of gas leak out during the first $8$ seconds?
If the tank starts out with $2$ gallons ( $256$ ounces) of gas, how much gas is in the tank after $3$ seconds? After $8$ seconds? Write an equation that can be used to calculate the amount of gas in the tank after $t$ seconds.
It takes Gary $1$ minute to realize what is going on. When does he have the most gas in the tank? Justify your answer.

7-89.

Differentiate each of the following functions. Homework Help ✎

$y =\frac { ( 3 x ^ { 2 } - 2 ) ^ { 3 } } { - 2 }$

$y = 6^{(3x^2 - 2)^3}$

$(2y)^2 − \cos(x^5) = 9$

7-90.

Integrate. Homework Help ✎

$\int _ { - 1 } ^ { 1 } 4 ^ { x + 2 } d x$

$\int 6t · 7^{3t^2}dt$

$\int$ $\sin^2(3m - 1) · \cos(3m - 1)dm$

7-91.

Write a possible equation for $y$ given $\frac { d y } { d x }= 12e^{4x} −\frac{1}{x}$ . Homework Help ✎

7-92.

Eddie is standing on third base, watching in amazement as Yuliana tries to steal second base. If Yuliana runs at $15$ ft/sec, how fast is her distance from Eddie changing when she is $40$ feet from first base? Homework Help ✎

Tilted Square, vertex on bottom, right vertex labeled first base, point on top right side labeled Yuliana, with arrow pointing towards top vertex, left vertex labeled Eddie, top left side labeled 90 feet, dashed segment between left vertex and right side midpoint

7-93.

Brooke and Anne came across this integral: $y =\int dx$ . Brooke says it cannot be evaluated because there is nothing to integrate. Anne says the answer is $y = x + C$ . Who is correct and why? Homework Help ✎

7-94.

Mr. Gauss is known for his indecisiveness! He will often walk down the hallway toward his office, change his mind, turn around, and then turn around again! One time he left class when the bell rang ( $t = 0$ ) traveling with velocity in feet per second given by $v(t) = 3t^2 - 17t + 10$ . After $7$ seconds, he arrived at his office. Homework Help ✎

Calculate the total distance he traveled during the $7$ seconds.
Assuming he traveled back and forth in a straight line, how far is his classroom from his office?

7-95.

Determine values for $a, b, c$ , and $d$ such that the function given at right is continuous and differentiable for all values of $x$ . Homework Help ✎

$f ( x ) = \left\{ \begin{array} { c c c } { 2 ^ { x } } & { \text { for } } & { x < 0 } \\ { a x ^ { 2 } + b x + c } & { \text { for } } & { 0 \leq x \leq 4 } \\ { d } & { \text { for } } & { x > 4 } \end{array} \right.$

Calculus Third Edition