{"id":1608,"date":"2020-04-04T10:02:36","date_gmt":"2020-04-04T10:02:36","guid":{"rendered":"http:\/\/ibalmaths.com\/?post_type=knowledgebase&#038;p=1608"},"modified":"2020-04-05T08:40:57","modified_gmt":"2020-04-05T08:40:57","slug":"ibdp-past-year-exam-questions-techniques-for-integration","status":"publish","type":"knowledgebase","link":"https:\/\/ibalmaths.com\/index.php\/ibdp-math-hl-2\/techniques-for-integration\/ibdp-past-year-exam-questions-techniques-for-integration\/","title":{"rendered":"IBDP Past Year Exam Questions &#8211; Techniques for Integration"},"content":{"rendered":"<p><strong>1. [M08\/P2\/TZ1]<\/strong><\/p>\r\n<p>By using an appropriate substitution find\u00a0<\/p>\r\n<p>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo>&#8747;<\/mo><mfrac><mrow><mi>tan<\/mi><mfenced><mrow><mi>ln<\/mi><mi>y<\/mi><\/mrow><\/mfenced><\/mrow><mi>y<\/mi><\/mfrac><mi>d<\/mi><mi>y<\/mi><mo>,<\/mo><mo>&#160;<\/mo><mo>&#160;<\/mo><mi>y<\/mi><mo>&#62;<\/mo><mn>0<\/mn><\/math>\r\n<\/p>\r\n<div id=\"link1-link-1608\" class=\"sh-link link1-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link1', 1608, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link1-toggle-1608\">Solution<\/span><\/a><\/div><div id=\"link1-content-1608\" class=\"sh-content link1-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1615\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-25.jpg\" alt=\"\" width=\"918\" height=\"538\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-25.jpg 918w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-25-300x176.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-25-768x450.jpg 768w\" sizes=\"auto, (max-width: 918px) 100vw, 918px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>2. [M98\/P1]<\/strong><\/p>\r\n<p>Find\u00a0 \u00a0\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo>&#8747;<\/mo><mi>a<\/mi><mi>r<\/mi><mi>c<\/mi><mi>tan<\/mi><mo>&#160;<\/mo><mi>x<\/mi><mo>&#160;<\/mo><mi>d<\/mi><mi>x<\/mi><\/math>\r\n .<\/p>\r\n<p><strong>3. [N95\/P1]<\/strong><\/p>\r\n<p>Find\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo>&#8747;<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><msup><mi>e<\/mi><mrow><mo>&#8211;<\/mo><mi>x<\/mi><\/mrow><\/msup><mo>&#160;<\/mo><mi>d<\/mi><mi>x<\/mi><\/math>\r\n .<\/p>\r\n<p><strong>4. [M11\/P2\/TZ2]<\/strong><\/p>\r\n<p>Using integration by parts, show that\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo>&#8747;<\/mo><msup><mi>e<\/mi><mrow><mn>2<\/mn><mi>x<\/mi><\/mrow><\/msup><mi>sin<\/mi><mi>x<\/mi><mo>&#160;<\/mo><mi>d<\/mi><mi>x<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>5<\/mn><\/mfrac><msup><mi>e<\/mi><mrow><mn>2<\/mn><mi>x<\/mi><\/mrow><\/msup><mfenced><mrow><mn>2<\/mn><mi>sin<\/mi><mi>x<\/mi><mo>&#160;<\/mo><mo>&#8211;<\/mo><mi>cos<\/mi><mi>x<\/mi><\/mrow><\/mfenced><mo>+<\/mo><mi>C<\/mi><\/math>\r\n . [6 marks]<\/p>\r\n<div id=\"link2-link-1608\" class=\"sh-link link2-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link2', 1608, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link2-toggle-1608\">Solution<\/span><\/a><\/div><div id=\"link2-content-1608\" class=\"sh-content link2-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1626\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-26.jpg\" alt=\"\" width=\"657\" height=\"202\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-26.jpg 657w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-26-300x92.jpg 300w\" sizes=\"auto, (max-width: 657px) 100vw, 657px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>5. [N12\/P2\/TZ2]<\/strong><\/p>\r\n<p>By using the substitution\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>x<\/mi><mo>=<\/mo><mi>sin<\/mi><mo>&#160;<\/mo><mi>t<\/mi><\/math>\r\n , find\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo>&#8747;<\/mo><mfrac><msup><mi>x<\/mi><mn>3<\/mn><\/msup><msqrt><mn>1<\/mn><mo>&#8211;<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><\/msqrt><\/mfrac><mi>d<\/mi><mi>x<\/mi><\/math>\r\n .\u00a0 \u00a0 [7 marks]<\/p>\r\n<div id=\"link5-link-1608\" class=\"sh-link link5-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link5', 1608, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link5-toggle-1608\">Solution<\/span><\/a><\/div><div id=\"link5-content-1608\" class=\"sh-content link5-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1628\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-27.jpg\" alt=\"\" width=\"935\" height=\"725\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-27.jpg 935w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-27-300x233.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-27-768x596.jpg 768w\" sizes=\"auto, (max-width: 935px) 100vw, 935px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>6. [M13\/P1\/TZ1]<\/strong><\/p>\r\n<p>The function\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>f<\/mi><\/math>\r\n is defined by\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>f<\/mi><mfenced><mi>x<\/mi><\/mfenced><mo>=<\/mo><mfrac><mn>1<\/mn><mrow><mn>4<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>&#8211;<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>5<\/mn><\/mrow><\/mfrac><\/math>\r\n . By using a suitable substitution show that\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo>&#8747;<\/mo><mi>f<\/mi><mfenced><mi>x<\/mi><\/mfenced><mo>&#160;<\/mo><mi>d<\/mi><mi>x<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>4<\/mn><\/mfrac><mo>&#8747;<\/mo><mfrac><mn>1<\/mn><mrow><msup><mi>u<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>1<\/mn><\/mrow><\/mfrac><mi>d<\/mi><mi>u<\/mi><\/math>\r\n .<\/p>\r\n<div id=\"link6-link-1608\" class=\"sh-link link6-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link6', 1608, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link6-toggle-1608\">Solution<\/span><\/a><\/div><div id=\"link6-content-1608\" class=\"sh-content link6-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1630\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-28.jpg\" alt=\"\" width=\"869\" height=\"319\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-28.jpg 869w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-28-300x110.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-28-768x282.jpg 768w\" sizes=\"auto, (max-width: 869px) 100vw, 869px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>7. [N13\/P2\/TZ0]<\/strong><\/p>\r\n<p>By using the substitution\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><mo>&#160;<\/mo><mi>tan<\/mi><mo>&#160;<\/mo><mi>u<\/mi><\/math>\r\n , show that\u00a0<\/p>\r\n<p>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo>&#8747;<\/mo><mfrac><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><msqrt><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>4<\/mn><\/msqrt><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mo>&#8211;<\/mo><msqrt><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>4<\/mn><\/msqrt><\/mrow><mrow><mn>4<\/mn><mi>x<\/mi><\/mrow><\/mfrac><mo>+<\/mo><mi>C<\/mi><\/math>\r\n \u00a0 \u00a0 \u00a0 \u00a0 [7 marks]<\/p>\r\n<div id=\"link7-link-1608\" class=\"sh-link link7-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link7', 1608, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link7-toggle-1608\">Solution<\/span><\/a><\/div><div id=\"link7-content-1608\" class=\"sh-content link7-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1631\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-29.jpg\" alt=\"\" width=\"967\" height=\"1194\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-29.jpg 967w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-29-243x300.jpg 243w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-29-829x1024.jpg 829w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-29-768x948.jpg 768w\" sizes=\"auto, (max-width: 967px) 100vw, 967px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>8. [N14\/P1\/TZ0]<\/strong><\/p>\r\n<p>By using the substitution\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>u<\/mi><mo>=<\/mo><mn>1<\/mn><mo>+<\/mo><msqrt><mi>x<\/mi><\/msqrt><\/math>\r\n , find\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo>&#8747;<\/mo><mfrac><msqrt><mi>x<\/mi><\/msqrt><mrow><mn>1<\/mn><mo>+<\/mo><msqrt><mi>x<\/mi><\/msqrt><\/mrow><\/mfrac><mo>&#160;<\/mo><mi>d<\/mi><mi>x<\/mi><\/math>\r\n .\u00a0 \u00a0[6 marks]<\/p>\r\n<div id=\"link1-link-1608\" class=\"sh-link link1-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link1', 1608, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link1-toggle-1608\">Solution<\/span><\/a><\/div><div id=\"link1-content-1608\" class=\"sh-content link1-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1638\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-32.jpg\" alt=\"\" width=\"1060\" height=\"544\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-32.jpg 1060w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-32-300x154.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-32-1024x526.jpg 1024w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-32-768x394.jpg 768w\" sizes=\"auto, (max-width: 1060px) 100vw, 1060px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>9. [M16\/P2\/TZ2]<\/strong><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1640\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-33.jpg\" alt=\"\" width=\"1114\" height=\"526\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-33.jpg 1114w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-33-300x142.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-33-1024x484.jpg 1024w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-33-768x363.jpg 768w\" sizes=\"auto, (max-width: 1114px) 100vw, 1114px\" \/><\/p>\r\n<div id=\"link9-link-1608\" class=\"sh-link link9-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link9', 1608, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link9-toggle-1608\">Solution<\/span><\/a><\/div><div id=\"link9-content-1608\" class=\"sh-content link9-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1641\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-34.jpg\" alt=\"\" width=\"997\" height=\"925\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-34.jpg 997w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-34-300x278.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-34-768x713.jpg 768w\" sizes=\"auto, (max-width: 997px) 100vw, 997px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>10. [N17\/P2\/TZ0]<\/strong><\/p>\r\n<p>By using the substitution\u00a0\r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mn>2<\/mn><mo>&#160;<\/mo><mi>s<\/mi><mi>e<\/mi><mi>c<\/mi><mo>&#160;<\/mo><mi>&#952;<\/mi><\/math>\r\n, show that\u00a0\r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo>&#8747;<\/mo><mfrac><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><mrow><mi>x<\/mi><msqrt><msup><mi>x<\/mi><mn>4<\/mn><\/msup><mo>&#8211;<\/mo><mn>4<\/mn><\/msqrt><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mn>1<\/mn><mn>4<\/mn><\/mfrac><mi>a<\/mi><mi>r<\/mi><mi>c<\/mi><mi>cos<\/mi><mfenced><mfrac><mn>2<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><\/mfrac><\/mfenced><mo>+<\/mo><mi>c<\/mi><\/math>\r\n.\u00a0 \u00a0 \u00a0[7 marks]<\/p>\r\n<div id=\"link10-link-1608\" class=\"sh-link link10-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link10', 1608, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link10-toggle-1608\">Solution<\/span><\/a><\/div><div id=\"link10-content-1608\" class=\"sh-content link10-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1643\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-35.jpg\" alt=\"\" width=\"1005\" height=\"1299\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-35.jpg 1005w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-35-232x300.jpg 232w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-35-792x1024.jpg 792w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-35-768x993.jpg 768w\" sizes=\"auto, (max-width: 1005px) 100vw, 1005px\" \/><\/p>\r\n<p>\u00a0<\/div><!-- AddThis Advanced Settings generic via filter on the_content --><!-- AddThis Share Buttons generic via filter on the_content -->","protected":false},"excerpt":{"rendered":"1. [M08\/P2\/TZ1] By using an appropriate substitution find\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 &#8747;tanlnyydy,&#160;&#160;y&#62;0 2. [M98\/P1] Find\u00a0 \u00a0\u00a0 &#8747;arctan&#160;x&#160;dx . 3. [N95\/P1] Find\u00a0 &#8747;x2e&#8211;x&#160;dx . 4. [M11\/P2\/TZ2] Using integration by parts, show that\u00a0 &#8747;e2xsinx&#160;dx=15e2x2sinx&#160;&#8211;cosx+C . [6 marks] 5. [N12\/P2\/TZ2] By using the substitution\u00a0 x=sin&#160;t , find\u00a0 &#8747;x31&#8211;x2dx .\u00a0 \u00a0 [7 marks] 6. [M13\/P1\/TZ1] [&hellip;]<!-- AddThis Advanced Settings generic via filter on get_the_excerpt --><!-- AddThis Share Buttons generic via filter on get_the_excerpt -->","protected":false},"author":4,"featured_media":0,"comment_status":"closed","ping_status":"closed","template":"","class_list":["post-1608","knowledgebase","type-knowledgebase","status-publish","hentry","knowledgebase_cat-techniques-for-integration","no-wpautop"],"_links":{"self":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase\/1608","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase"}],"about":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/types\/knowledgebase"}],"author":[{"embeddable":true,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/users\/4"}],"replies":[{"embeddable":true,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/comments?post=1608"}],"version-history":[{"count":16,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase\/1608\/revisions"}],"predecessor-version":[{"id":1644,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase\/1608\/revisions\/1644"}],"wp:attachment":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/media?parent=1608"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}