Exam 5: Integrals

Determine a reduction formula for $\int _ { 1 } ^ { e } x ( \ln x ) ^ { n } d x$

Use FTC1 to write down an antiderivative $F ( x ) \text { of } f ( x ) = e ^ { x ^ { 2 } } \text { such that } F ( 0 ) = 2$

Let $g ( x ) = \int _ { - 2 } ^ { x } h ( t ) d t$ , where h is the function whose graph is shown below: (a) Evaluate g ( $- 2$ ), g ( $2$ ), g (0), g (3).(b) On what interval(s) is g increasing? (c) What are the maximum and minimum values of g over [ $- 2$ ,3]?

Use the definition of the definite integral to evaluate $\int _ { 0 } ^ { 4 } \left( 2 x - x ^ { 2 } \right) d x$

Evaluate the improper integral $\int _ { 1 } ^ { \infty } \frac { \ln x } { x } d x$

Use Simpson's Rule with n = 4 to approximate $\int _ { 0 } ^ { 1 } \frac { 1 } { 1 + x ^ { 2 } } d x$

Express $\int _ { 1 } ^ { 4 } \left( 2 x - \frac { 1 } { x ^ { 2 } } \right) d x$ as the limit of a sum.

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.(a) $\int _ { 0 } ^ { 2 } \frac { 1 } { x - 1 } d x$ (b) $\int _ { 2 } ^ { \infty } \frac { 1 } { x ( \ln x ) ^ { 2 } } d x$ (c) $\int _ { 0 } ^ { 1 } \frac { \ln x } { \sqrt { x } } d x$

Use the definition of the definite integral to evaluate $\int _ { 0 } ^ { 1 } ( 1 - 2 x ) d x$

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

Use FTC1 to write down an antiderivative $F ( x ) \text { of } f ( x ) = \sqrt { x } e ^ { x } \text { such that } F ( 1 ) = 4$

Let $s ( t ) = \int _ { 1 } ^ { t ^ { 2 } - 3 t } ( u + 2 ) e ^ { \sqrt [ 4 ] { u } } d u$ be the position function of a particle. When does the particle stop?

Evaluate $\int _ { - 2 } ^ { 2 } x \cos \left( x ^ { 3 } \right) d x$

Use the Comparison Theorem to determine whether $\int _ { 2 } ^ { \infty } \frac { d x } { \sqrt { 4 x ^ { 2 } + 1 } }$ is convergent or divergent. Justify your answer.

Find the value of the integral $\int _ { 0 } ^ { 1 } \sqrt { 1 - u ^ { 2 } } d u$

Given $\int _ { 0 } ^ { 4 } f ( x ) d x = - 5 \text { and } \int _ { 0 } ^ { 6 } f ( x ) d x = 3$ use the properties of definite integrals to find $\int _ { 6 } ^ { 4 } f ( x ) d x .$

If $\int _ { 1 } ^ { 2 x } f ( t ) d t = x \sin ^ { - 1 } x$ where f is a continuous function, find f (1).

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 right endpoint of the subinterval.

Find the value of the integral $\int _ { 1 } ^ { 3 } \frac { 1 } { x ^ { 2 } } d x$