⧼exchistory⧽
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BBy Bot
Nov 03'24
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  • lab{7.5.1a} Integrate [math]\int \sec x \; dx[/math] by the technique for integrating rational functions of trigonometric functions.
  • We have already shown (see \ref{thm 7.2.3}) that
    [[math]] \int \sec x \; dx = \ln|\sec x + \tan x| + c . [[/math]]
    Show that this solution agrees with the one found in \ref{ex7.5.1a} for an appropriate choice of the constant [math]c[/math].
BBy Bot
Nov 03'24
[math] \newcommand{\ex}[1]{\item } \newcommand{\sx}{\item} \newcommand{\x}{\sx} \newcommand{\sxlab}[1]{} \newcommand{\xlab}{\sxlab} \newcommand{\prov}[1] {\quad #1} \newcommand{\provx}[1] {\quad \mbox{#1}} \newcommand{\intext}[1]{\quad \mbox{#1} \quad} \newcommand{\R}{\mathrm{\bf R}} \newcommand{\Q}{\mathrm{\bf Q}} \newcommand{\Z}{\mathrm{\bf Z}} \newcommand{\C}{\mathrm{\bf C}} \newcommand{\dt}{\textbf} \newcommand{\goesto}{\rightarrow} \newcommand{\ddxof}[1]{\frac{d #1}{d x}} \newcommand{\ddx}{\frac{d}{dx}} \newcommand{\ddt}{\frac{d}{dt}} \newcommand{\dydx}{\ddxof y} \newcommand{\nxder}[3]{\frac{d^{#1}{#2}}{d{#3}^{#1}}} \newcommand{\deriv}[2]{\frac{d^{#1}{#2}}{dx^{#1}}} \newcommand{\dist}{\mathrm{distance}} \newcommand{\arccot}{\mathrm{arccot\:}} \newcommand{\arccsc}{\mathrm{arccsc\:}} \newcommand{\arcsec}{\mathrm{arcsec\:}} \newcommand{\arctanh}{\mathrm{arctanh\:}} \newcommand{\arcsinh}{\mathrm{arcsinh\:}} \newcommand{\arccosh}{\mathrm{arccosh\:}} \newcommand{\sech}{\mathrm{sech\:}} \newcommand{\csch}{\mathrm{csch\:}} \newcommand{\conj}[1]{\overline{#1}} \newcommand{\mathds}{\mathbb} [/math]
  • lab{7.5.2a} Integrate [math]\int \csc x \; dx[/math] by the technique for integrating rational functions of trigonometric functions.
  • The formula [math]\int \csc x \; dx = -\ln|\csc x+\cot x|+c[/math] is given in Problem. Show that this integral agrees with the one obtained in \ref{ex7.5.2a} for an appropriate choice of [math]c[/math].
BBy Bot
Nov 03'24
[math] \newcommand{\ex}[1]{\item } \newcommand{\sx}{\item} \newcommand{\x}{\sx} \newcommand{\sxlab}[1]{} \newcommand{\xlab}{\sxlab} \newcommand{\prov}[1] {\quad #1} \newcommand{\provx}[1] {\quad \mbox{#1}} \newcommand{\intext}[1]{\quad \mbox{#1} \quad} \newcommand{\R}{\mathrm{\bf R}} \newcommand{\Q}{\mathrm{\bf Q}} \newcommand{\Z}{\mathrm{\bf Z}} \newcommand{\C}{\mathrm{\bf C}} \newcommand{\dt}{\textbf} \newcommand{\goesto}{\rightarrow} \newcommand{\ddxof}[1]{\frac{d #1}{d x}} \newcommand{\ddx}{\frac{d}{dx}} \newcommand{\ddt}{\frac{d}{dt}} \newcommand{\dydx}{\ddxof y} \newcommand{\nxder}[3]{\frac{d^{#1}{#2}}{d{#3}^{#1}}} \newcommand{\deriv}[2]{\frac{d^{#1}{#2}}{dx^{#1}}} \newcommand{\dist}{\mathrm{distance}} \newcommand{\arccot}{\mathrm{arccot\:}} \newcommand{\arccsc}{\mathrm{arccsc\:}} \newcommand{\arcsec}{\mathrm{arcsec\:}} \newcommand{\arctanh}{\mathrm{arctanh\:}} \newcommand{\arcsinh}{\mathrm{arcsinh\:}} \newcommand{\arccosh}{\mathrm{arccosh\:}} \newcommand{\sech}{\mathrm{sech\:}} \newcommand{\csch}{\mathrm{csch\:}} \newcommand{\conj}[1]{\overline{#1}} \newcommand{\mathds}{\mathbb} [/math]

Integrate each of the following.

  • [math]\int \frac{x\;dx}{\sqrt[3]{1+x}}[/math]
  • [math]\int \frac{x\;dx}{1+\sqrt[3]x}[/math]
  • [math]\int \frac{\sin x\;dx}{1+\sin x}[/math]
  • [math]\int \frac{3\;dx}{\sin x + \cos x}[/math]
  • [math]\int \frac{dt}{2+\cos t}[/math]
  • [math]\int \frac{(1+x)^{\frac15}}{(1+x)^{\frac13}}\;dx[/math]
  • [math]\int \frac{dy}{\sqrt y + \sqrt[3]y}[/math]
  • [math]\int \frac{\tan x \; dx}{1+\tan^2x}[/math]
  • [math]\int \frac{x^2}{\sqrt{5x+3}}\;dx[/math]
  • [math]\int \frac{dx}{(1+\sqrt x)^5}[/math]
  • [math]\int \frac{e^x\;dx}{\sqrt{1+e^x}}[/math]
  • [math]\int \frac{dx}{\sqrt{1+e^x}}[/math].
BBy Bot
Nov 03'24
[math] \newcommand{\ex}[1]{\item } \newcommand{\sx}{\item} \newcommand{\x}{\sx} \newcommand{\sxlab}[1]{} \newcommand{\xlab}{\sxlab} \newcommand{\prov}[1] {\quad #1} \newcommand{\provx}[1] {\quad \mbox{#1}} \newcommand{\intext}[1]{\quad \mbox{#1} \quad} \newcommand{\R}{\mathrm{\bf R}} \newcommand{\Q}{\mathrm{\bf Q}} \newcommand{\Z}{\mathrm{\bf Z}} \newcommand{\C}{\mathrm{\bf C}} \newcommand{\dt}{\textbf} \newcommand{\goesto}{\rightarrow} \newcommand{\ddxof}[1]{\frac{d #1}{d x}} \newcommand{\ddx}{\frac{d}{dx}} \newcommand{\ddt}{\frac{d}{dt}} \newcommand{\dydx}{\ddxof y} \newcommand{\nxder}[3]{\frac{d^{#1}{#2}}{d{#3}^{#1}}} \newcommand{\deriv}[2]{\frac{d^{#1}{#2}}{dx^{#1}}} \newcommand{\dist}{\mathrm{distance}} \newcommand{\arccot}{\mathrm{arccot\:}} \newcommand{\arccsc}{\mathrm{arccsc\:}} \newcommand{\arcsec}{\mathrm{arcsec\:}} \newcommand{\arctanh}{\mathrm{arctanh\:}} \newcommand{\arcsinh}{\mathrm{arcsinh\:}} \newcommand{\arccosh}{\mathrm{arccosh\:}} \newcommand{\sech}{\mathrm{sech\:}} \newcommand{\csch}{\mathrm{csch\:}} \newcommand{\conj}[1]{\overline{#1}} \newcommand{\mathds}{\mathbb} [/math]

Evaluate each of the following definite integrals.

  • [math]\int_{\frac{\pi}6}^{\frac{\pi}3} \frac{dx}{\cos^2x\sin x}[/math]
  • [math]\int_0^3 \frac{x\;dx}{\sqrt{1+x}}[/math]
  • [math]\int_0^{\frac{\pi}2} \sin^5x \cos^2x \; dx[/math]
  • [math]\int_1^{64} \frac{dx}{\sqrt[3]x + 2\sqrt{x}}[/math].