Exam 5: Integrals

Use the Table of Integrals in your textbook to evaluate each of the following: (a) $\int e ^ { 3 x } \cos 2 x d x$ (b) $\int \sqrt { 4 x - x ^ { 2 } } d x$ (c) $\int \sqrt { 4 x - 25 x ^ { 2 } } d x$ (d) $\int u ^ { 3 } \sin u d u$ (e) $\int \cos ^ { 4 } ( 2 t ) d t$ (f) $\int \frac { 1 } { x ^ { 2 } + 2 x + 2 } d x$ (g) $\int \sqrt { 9 + 4 u ^ { 2 } } d u$ (h) $\int \sqrt { 9 - 4 u ^ { 2 } } d u$ (i) $\int \sqrt { 4 u ^ { 2 } - 9 } d u$ (j) $\int \sin ^ { 2 } t \cos t e ^ { \sin t } d t$

Find the value of the integral $\int _ { 1 } ^ { 4 } \frac { 1 } { ( 1 + \sqrt { x } ) ^ { 2 } } \frac { 1 } { \sqrt { x } } d x$

Evaluate the improper integral $\int _ { 0 } ^ { 4 } \frac { 1 } { 4 - x } d x$

Evaluate the improper integral $\int _ { 0 } ^ { \infty } \cos x d x$

The graph of a function f is given below. List the following quantities from smallest to largest. (a) $\int _ { 0 } ^ { 9 } f ( x ) d x$ (b) $\int _ { 6 } ^ { 9 } f ( x ) d x$ (c) $\int _ { 0 } ^ { 5 } f ( x ) d x$ (d) $\int _ { 0 } ^ { 6 } f ( x ) d x$ (e) $f ^ { \prime } ( 5 )$

Use Simpson's Rule with n = 2 to approximate the integral $\int _ { 0 } ^ { 1 } x ^ { 3 } d x$

Find the value of the integral $\int _ { 0 } ^ { 8 } \sqrt [ 3 ] { x } \left( \frac { 1 } { \sqrt [ 3 ] { x ^ { 2 } } } + 1 \right) d x$

Which one of the following integrals is improper?

Find the value of the integral $\int _ { 0 } ^ { 1 } \frac { x ^ { 2 } } { \left( x ^ { 3 } + 1 \right) ^ { 2 } } d x$

A dry ice puck is pushed across an uneven surface. Below is a graph of the velocity v (in cm/s) as a function of the time t (in seconds). Determine the total distance traveled by the puck for $0 \leq t \leq 9$ .

Find the value of the integral $\int _ { - 3 } ^ { 3 } \sqrt [ 3 ] { 2 x ^ { 3 } - 5 x } d x$

Find the value of the integral $\int _ { 4 - \sqrt { 3 } } ^ { 4 + \sqrt { 3 } } \frac { 1 } { x ^ { 2 } - 8 x + 17 } d x$

Find the value of the integral $\int _ { 0 } ^ { \sqrt { 3 } / 2 } \frac { 1 } { 4 \sqrt { 1 - t ^ { 2 } } } d t$

Find the partial fraction expansion of the rational function: $\frac { x } { x ^ { 2 } - 5 x + 6 }$ .

Given the graph below, use 5 rectangles to find an estimate for the area under the graph from x = 0 to x = 10. Find lower and upper estimates for the area.

Let f (x) = $x ^ { 2 } - 2$ on the interval [0,2]. Let the interval be divided into four equal subintervals. Find the value of the Riemann sum $\sum _ { i = 1 } ^ { 4 } f \left( x _ { i } ^ { * } \right) \Delta x _ { i } \text { if } x _ { i } ^ { * }$ is the left endpoint of the subinterval.

Evaluate the following integrals: (a) $\int x e ^ { x ^ { 2 } } d x$ (b) $\int \frac { \cos ( \ln x ) } { x } d x$ (c) $\int \frac { \sec ^ { 2 } x } { \tan x } d x$ (d) $\int \frac { 3 x ^ { 2 } } { 1 + x ^ { 6 } } d x$

Compute $\frac { d } { d x } \int _ { \cos x } ^ { x ^ { 3 } } e ^ { - t ^ { 2 } } d t$

Let $g ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$ where f is the function whose graph is shown below: (a) Use the Comparison Properties of Integrals to estimate the value of the following integrals: (i) $\int _ { 0 } ^ { 3 } f ( t ) d t$ (ii) $\int _ { 3 } ^ { 6 } f ( t ) d t$ (iii) $\int _ { 6 } ^ { 9 } f ( t ) d t$ (iv) $\int _ { 9 } ^ { 10 } f ( t ) d t$ (b) At what values of x does g have a local maximum or minimum? (c) At what values of x does g attain its absolute maximum or minimum?

Compute $\frac { d } { d x } \int _ { 2 } ^ { e ^ { x ^ { 2 } } } \ln ( t ) d t$