DP Mathematics HL Questionbank

• Syllabus

6.7

Sub sections and their related questions

Integration by substitution.

• 12M.1.hl.TZ1.12c:  Use the substitution \(x = {\sin ^2}\theta \) to show that...
• 12N.2.hl.TZ0.8: By using the substitution \(x = \sin t\) , find...
• 08M.2.hl.TZ1.9: By using an appropriate substitution...
• 11M.1.hl.TZ2.13b: Find the value of \(\int_0^1 {\sqrt {\frac{x}{{4 - x}}} }{{\text{d}}x} \) using the substitution...
• 11M.3ca.hl.TZ0.4b: By letting \(y = x - n\pi \) , show that \({I_n} = {{\text{e}}^{ - n\pi }}{I_0}\) .
• 09N.1.hl.TZ0.7a: Calculate...
• SPNone.1.hl.TZ0.11a: Find the value of the integral \(\int_0^{\sqrt 2 } {\sqrt {4 - {x^2}} {\text{d}}x} \) .
• SPNone.1.hl.TZ0.11c: Using the substitution \(t = \tan \theta \) , find the value of the...
• 13M.1.hl.TZ1.12e: By using a suitable substitution show that...
• 10M.1.hl.TZ1.9: (a)     Given that \(\alpha  > 1\), use the substitution \(u = \frac{1}{x}\) to show...
• 09M.2.hl.TZ1.6: (a)     Integrate \(\int {\frac{{\sin \theta }}{{1 - \cos \theta }}} {\text{d}}\theta \)...
• 09M.2.hl.TZ2.9: Using the substitution \(x = 2\sin \theta \) , show...
• 14M.2.hl.TZ1.6b: Find \(\int {f(x){\text{d}}x} \).
• 14M.1.hl.TZ2.10: Use the substitution \(x = a\sec \theta \) to show that...
• 14M.2.hl.TZ2.14b: Use the substitution \(u = {t^2}\) to find \(\int {\frac{t}{{12 + {t^4}}}{\text{d}}t} \).
• 14N.1.hl.TZ0.6: By using the substitution \(u = 1 + \sqrt x \), find...
• 15M.1.hl.TZ1.8: By using the substitution \(u = {{\text{e}}^x} + 3\), find...
• 15M.1.hl.TZ2.8: By using the substitution \(t = \tan x\), find...
• 15N.1.hl.TZ0.5: Use the substitution \(u = \ln x\) to find the value of...
• 15N.1.hl.TZ0.12f: Find the area of the region enclosed by the graph of \(y = f(x)\) and the \(x\)-axis for...
• 16M.1.hl.TZ1.13b: Use the substitution \(u = \ln x\) to find the area of the region \(R\).
• 16M.1.hl.TZ1.13c: (i)     Find the value of \({I_0}\). (ii)     Prove that...
• 17M.1.hl.TZ2.6a: Using the substitution \(x = \tan \theta \) show that...
• 17M.1.hl.TZ2.6b: Hence find the value of...
• 17N.2.hl.TZ0.8: By using the substitution \({x^2} = 2\sec \theta \), show that...
• 18M.1.hl.TZ2.8a: Use the substitution \(u = {x^{\frac{1}{2}}}\) to...
• 18M.1.hl.TZ2.8b: Hence find the value...

Integration by parts.

• 12M.1.hl.TZ2.10c: The region bounded by the graph, the x-axis and the y-axis is denoted by A and the region bounded...
• 08M.1.hl.TZ2.6: Show that...
• 08N.1.hl.TZ0.5: Calculate the exact value of \(\int_1^{\text{e}} {{x^2}\ln x{\text{d}}x} \) .
• 11M.2.hl.TZ2.13B: (a)     Using integration by parts, show that...
• 11M.3ca.hl.TZ0.4a: Show that \({I_0} = \frac{1}{2}(1 + {{\text{e}}^{ - \pi }})\) .
• SPNone.1.hl.TZ0.11b: Find the value of the integral \(\int_0^{0.5} {\arcsin x {\text{d}}x} \) .
• 10M.1.hl.TZ2.9: Find the value of \(\int_0^1 {t\ln (t + 1){\text{d}}t} \).
• 13M.2.hl.TZ2.4a: Find \(\int {x{{\sec }^2}x{\text{d}}x} \).
• 15M.1.hl.TZ2.5: Show that \(\int_1^2 {{x^3}\ln x{\text{d}}x = 4\ln 2 - \frac{{15}}{{16}}} \).
• 15N.1.hl.TZ0.2: Using integration by parts find \(\int {x\sin x{\text{d}}x} \).
• 16N.1.hl.TZ0.11f: Find the area of the region enclosed by the graph of \(f\) and the \(x\)-axis. The curvature at...
• 17M.1.hl.TZ1.9: Find \(\int {\arcsin x\,{\text{d}}x} \)
• 17M.1.hl.TZ2.9b: Find \(\int {f(x)\cos x{\text{d}}x} \).
• 18M.1.hl.TZ2.6b: Hence, or otherwise, find...