적분 기법 전체 문제

The Laplace transform of a continuous function over the interval $[0,\infty )$ is defined by $F(s)={\int }_{0}^{\infty }{e}^{-sx}f(x)dx$ (see the Student Project). This definition is used to solve some important initial-value problems in differential equations, as discussed later. The domain of F is the set of all real numbers s such that the improper integral converges. Find the Laplace transform F of each of the following functions and give the domain of F.

$f(x)={e}^{ax}$

Use the formula for arc length to show that the circumference of the circle ${x}^{2}+{y}^{2}=1$ is $2\pi .$

A non-negative function is a probability density function if it satisfies the following definition: ${\int }_{-\infty }^{\infty }f(t)dt=1.$ The probability that a random variable x lies between a and b is given by $P(a\le x\le b)={\int }_{a}^{b}f(t)dt.$

Show that $f(x)=\{\begin{array}{c}0\text{if}x<0 \\ 7{e}^{-7x}\text{if}x\ge 0\end{array}$ is a probability density function.

Find the probability that x is between 0 and 0.3. (Use the function defined in the preceding problem.) Use four-place decimal accuracy.

Chapter Review Exercises

For the following exercises, determine whether the statement is true or false. Justify your answer with a proof or a counterexample.

$\int {e}^{x}\sin (x)dx$ cannot be integrated by parts.

Chapter Review Exercises

For the following exercises, determine whether the statement is true or false. Justify your answer with a proof or a counterexample.

$\int \frac{1}{{x}^{4}+1}dx$ cannot be integrated using partial fractions.

In numerical integration, increasing the number of points decreases the error.

Integration by parts can always yield the integral.

For the following exercises, evaluate the integral using the specified method.

$\int {x}^{2}\sin (4x)dx$ using integration by parts

For the following exercises, evaluate the integral using the specified method.

$\int \frac{1}{{x}^{2}\sqrt{{x}^{2}+16}}dx$ using trigonometric substitution

For the following exercises, evaluate the integral using the specified method.

$\int \sqrt{x}\ln (x)dx$ using integration by parts

For the following exercises, evaluate the integral using the specified method.

$\int \frac{3x}{{x}^{3}+2{x}^{2}-5x-6}dx$ using partial fractions

For the following exercises, evaluate the integral using the specified method.

$\int \frac{{x}^{5}}{{(4{x}^{2}+4)}^{5/2}}dx$ using trigonometric substitution

For the following exercises, evaluate the integral using the specified method.

$\int \frac{\sqrt{4-{\sin }^{2}(x)}}{{\sin }^{2}(x)}\cos (x)dx$ using a table of integrals or a CAS

For the following exercises, integrate using whatever method you choose.

$\int {\sin }^{2}(x){\cos }^{2}(x)dx$

For the following exercises, integrate using whatever method you choose.

$\int {x}^{3}\sqrt{{x}^{2}+2}dx$

For the following exercises, integrate using whatever method you choose.

$\int \frac{3{x}^{2}+1}{{x}^{4}-2{x}^{3}-{x}^{2}+2x}dx$

For the following exercises, integrate using whatever method you choose.

$\int \frac{1}{{x}^{4}+4}dx$

For the following exercises, integrate using whatever method you choose.

$\int \frac{\sqrt{3+16{x}^{4}}}{{x}^{3}}dx$

For the following exercises, approximate the integrals using the midpoint rule, trapezoidal rule, and Simpson’s rule using four subintervals, rounding to three decimals.

[T] ${\int }_{1}^{2}\sqrt{{x}^{5}+2}dx$