{"id":20183,"date":"2024-04-15T05:49:29","date_gmt":"2024-04-15T05:49:29","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=20183"},"modified":"2024-04-15T05:49:29","modified_gmt":"2024-04-15T05:49:29","slug":"the-value-of-the-integral-intlimits_02-intlimits_0textx-textetextx-texty-textdy-dx-a-frac12left-texte-1-right-b","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/the-value-of-the-integral-intlimits_02-intlimits_0textx-textetextx-texty-textdy-dx-a-frac12left-texte-1-right-b\/","title":{"rendered":"The value of the integral \\[\\int\\limits_0^2 {\\int\\limits_0^{\\text{x}} {{{\\text{e}}^{{\\text{x}} + {\\text{y}}}}} } {\\text{dy dx}}\\] A. \\[\\frac{1}{2}\\left( {{\\text{e}} &#8211; 1} \\right)\\] B. \\[\\frac{1}{2}{\\left( {{{\\text{e}}^2} &#8211; 1} \\right)^2}\\] C. \\[\\frac{1}{2}\\left( {{{\\text{e}}^2} &#8211; {\\text{e}}} \\right)\\] D. \\[\\frac{1}{2}{\\left( {{\\text{e}} &#8211; \\frac{1}{{\\text{e}}}} \\right)^2}\\]"},"content":{"rendered":"<p>[amp_mcq option1=&#8221;\\[\\frac{1}{2}\\left( {{\\text{e}} &#8211; 1} \\right)\\]&#8221; option2=&#8221;\\[\\frac{1}{2}{\\left( {{{\\text{e}}^2} &#8211; 1} \\right)^2}\\]&#8221; option3=&#8221;\\[\\frac{1}{2}\\left( {{{\\text{e}}^2} &#8211; {\\text{e}}} \\right)\\]&#8221; option4=&#8221;\\[\\frac{1}{2}{\\left( {{\\text{e}} &#8211; \\frac{1}{{\\text{e}}}} \\right)^2}\\]&#8221; correct=&#8221;option1&#8243;]<!--more--><\/p>\n<p>The correct answer is $\\boxed{\\frac{1}{2}{\\left( {{{\\text{e}}^2} &#8211; 1} \\right)^2}}$.<\/p>\n<p>To evaluate the integral, we can substitute in the limits of integration. We get:<\/p>\n<p>$$\\int_0^2 {\\int_0^{\\text{x}} {{{\\text{e}}^{{\\text{x}} + {\\text{y}}}}} } {\\text{dy dx}} = \\int_0^2 {\\text{e}^{\\text{x}} \\int_0^{\\text{x}} {{{\\text{e}}^{\\text{y}}}}} } {\\text{dy dx}} = \\int_0^2 {\\text{e}^{\\text{x}} \\left[ {{\\text{e}}^{\\text{y}}} \\right]_0^{\\text{x}} } {\\text{dx}} = \\int_0^2 {\\text{e}^{2\\text{x}}} {\\text{dx}}$$<\/p>\n<p>We can then evaluate the integral using the following formula:<\/p>\n<p>$$\\int {\\text{e}^{kx}} {\\text{dx}} = \\frac{\\text{e}^{kx}}{k} + C$$<\/p>\n<p>where $C$ is an arbitrary constant.<\/p>\n<p>We get:<\/p>\n<p>$$\\int_0^2 {\\text{e}^{2\\text{x}}} {\\text{dx}} = \\frac{\\text{e}^{2\\text{x}}}{2} \\bigg|_0^2 = \\frac{\\text{e}^{4} &#8211; \\text{e}^0}{2} = \\frac{1}{2}{\\left( {{{\\text{e}}^2} &#8211; 1} \\right)^2}$$<\/p>\n","protected":false},"excerpt":{"rendered":"<p>[amp_mcq option1=&#8221;\\[\\frac{1}{2}\\left( {{\\text{e}} &#8211; 1} \\right)\\]&#8221; option2=&#8221;\\[\\frac{1}{2}{\\left( {{{\\text{e}}^2} &#8211; 1} \\right)^2}\\]&#8221; option3=&#8221;\\[\\frac{1}{2}\\left( {{{\\text{e}}^2} &#8211; {\\text{e}}} \\right)\\]&#8221; option4=&#8221;\\[\\frac{1}{2}{\\left( {{\\text{e}} &#8211; \\frac{1}{{\\text{e}}}} \\right)^2}\\]&#8221; correct=&#8221;option1&#8243;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[690],"tags":[],"class_list":["post-20183","post","type-post","status-publish","format-standard","hentry","category-calculus","no-featured-image-padding"],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v22.2 (Yoast SEO v23.3) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>The value of the integral \\[\\int\\limits_0^2 {\\int\\limits_0^{\\text{x}} {{{\\text{e}}^{{\\text{x}} + {\\text{y}}}}} } {\\text{dy dx}}\\] A. \\[\\frac{1}{2}\\left( {{\\text{e}} - 1} \\right)\\] B. \\[\\frac{1}{2}{\\left( {{{\\text{e}}^2} - 1} \\right)^2}\\] C. \\[\\frac{1}{2}\\left( {{{\\text{e}}^2} - {\\text{e}}} \\right)\\] D. \\[\\frac{1}{2}{\\left( {{\\text{e}} - \\frac{1}{{\\text{e}}}} \\right)^2}\\]<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/exam.pscnotes.com\/mcq\/the-value-of-the-integral-intlimits_02-intlimits_0textx-textetextx-texty-textdy-dx-a-frac12left-texte-1-right-b\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"The value of the integral \\[\\int\\limits_0^2 {\\int\\limits_0^{\\text{x}} {{{\\text{e}}^{{\\text{x}} + {\\text{y}}}}} } {\\text{dy dx}}\\] A. \\[\\frac{1}{2}\\left( {{\\text{e}} - 1} \\right)\\] B. \\[\\frac{1}{2}{\\left( {{{\\text{e}}^2} - 1} \\right)^2}\\] C. \\[\\frac{1}{2}\\left( {{{\\text{e}}^2} - {\\text{e}}} \\right)\\] D. \\[\\frac{1}{2}{\\left( {{\\text{e}} - \\frac{1}{{\\text{e}}}} \\right)^2}\\]\" \/>\n<meta property=\"og:description\" content=\"[amp_mcq option1=&#8221;[frac{1}{2}left( {{text{e}} &#8211; 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