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...