Integral Calculus Classwork

math-1109 MATH-1109 Differential and Integral Calculus This comprehensive list categorizes the mathematical problems found in your sources, distinguishing between those that are solved or partially worked out, and those that are listed primarily for practice.

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I. Indefinite Integrals (Total: 49 problems)

This section includes specific examples and long lists of practice integrals based on various substitution techniques.

A. Solved / Worked Out Examples (9 problems)

Problem	Status
$\int \frac{dx}{\sqrt{(x-\alpha )(\beta -x)}}$	Solved (General Form)
$\int \frac{dx}{x+2\sqrt{x+3}}$	Solved
$\int \frac{2x+7}{\sqrt{x^2-2x+8}}dx$	Solved
$\int \frac{x-1}{x\sqrt{x^2+1}}dx$	Solved (via substitution $x=1/z$)
$\int \frac{dx}{2x+3\sqrt{x^2+3x+2}}$	Solved (via substitution $2x+3=1/z$)
$\int \frac{dx}{x^2+1\sqrt{x^2+4}}$	Solved (via substitutions $x=1/z$ and $1+4z^2=t^2$)
$\int \frac{x^2+x+1}{\sqrt{x^2+2x+3}}dx$	Solved (Decomposition Setup shown)
$\int \frac{dx}{5-13\sin x}$	Solved (via half-angle tangent substitution)
$\int \frac{2\sin x+3\cos x}{3\sin x+4\cos x}dx$	Solved (Coefficient setup and final integral shown)

B. Unsolved Practice Problems (40 problems)

Problem Type	List of Unsolved Integrals
Forms with Square Roots (Rational Denominators)	$\int \frac{dx}{\sqrt{(x-\alpha )(x-\beta )}}$, $\int \frac{dx}{\sqrt{(\alpha -x)(x-\beta )}}$, $\int \frac{dx}{(x+2)\sqrt{x+3}}$, $\int \frac{dx}{(x-3)\sqrt{x-2}}$, $\int \frac{dx}{2x+1\sqrt{4x+3}}$, $\int \frac{dx}{2x\sqrt{x+1}}$, $\int \frac{dx}{x+2\sqrt{x+3}}$, $\int \frac{dx}{1-\sqrt{x}}$
Forms $\int \frac{Px+Q}{\sqrt{A{x}^2+Bx+C}}dx$	$\int \frac{7x-9}{\sqrt{x^2-2x+35}}$, $\int \frac{x+1}{\sqrt{x^2+4x+5}}$, $\int \frac{x+1}{\sqrt{3+2x-x^2}}$, $\int \frac{2x+5}{\sqrt{x^2+x+1}}$, $\int \frac{x^2-2x+5}{\sqrt{2x^2-8x+5}}$
Forms $\int \frac{Px+Q}{A{x}^2+Bx+C}dx$	$\int \frac{2x+3}{2x^2+2x+1}$, $\int \frac{4x+3}{3x^2+3x+1}$, $\int \frac{x+b}{\sqrt{x^2+a^2}}$
Forms $\int \frac{dx}{f(x)\sqrt{g(x)}}$	$\int \frac{dx}{(2x+1)\sqrt{4x+3}}$, $\int \frac{dx}{x\sqrt{x^2+1}}$, $\int \frac{dx}{x\sqrt{1-x^2}}$, $\int \frac{dx}{1+x\sqrt{1-x^2}}$, $\int \frac{dx}{x\sqrt{x^2+2x}}$, $\int \frac{dx}{1+x^2\sqrt{1-x^2}}$, $\int \frac{dx}{x^2\sqrt{1-x^2}}$, $\int \frac{dx}{x^2\sqrt{x^2-1}}$
Integrals involving $\sqrt{Q}$	$\int (3x-2)\sqrt{x^2-x+1}dx$ (Unfinished), $\int \sqrt{(x-\alpha )(\beta -x)}dx$, $\int (x-1)\sqrt{x^2-1}dx$, $\int (x+b)\sqrt{x^2+b^2}dx$, $\int (x+2)\sqrt{2x^2+2x+1}dx$, $\int \frac{x^2+x+1}{\sqrt{x^2+1}}dx$, $\int \frac{x^2+2x+3}{\sqrt{x^2+2x+1}}dx$, $\int \frac{x^3+2x^2+x-3}{\sqrt{x^2+2x+3}}dx$, $\int x\sqrt{\frac{a-x}{a+x}}dx$
Trigonometric Integrals	$\int \frac{dx}{13+3\cos x+4\sin x}$, $\int \frac{dx}{a+b\sin x}$, $\int \frac{dx}{5+4\sin x}$, $\int \frac{dx}{5+4\cos x}$, $\int \frac{dx}{3+{\sec }^{2}x}$, $\int \frac{dx}{a^2{\cos }^{2}x-b^2{\sin }^{2}x}$, $\int \frac{\sin xdx}{\sqrt{a^2{\cos }^{2}x+b^2{\sin }^{2}x}}$, $\int \frac{\cos xdx}{2\sin x+3\cos x}$, $\int \frac{dx}{1+\sqrt{\tan x}}$, $\int \frac{11\cos x-16\sin x}{20\sin x+5\cos x}$

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II. Definite Integrals (Total: 19 problems)

A. Solved / Worked Out Examples (6 problems)

Problem	Status
${\int }_{\alpha }^{\beta }\frac{dx}{\sqrt{(\beta -x)(x-\alpha )}}$	Solved (Evaluated to $\pi$)
${\int }_{\alpha }^{\beta }\sqrt{(x-\alpha )(\beta -x)}dx$	Solved (Setup shown)
${\int }_0^{\pi /2}\frac{dx}{a^2{\cos }^{2}x+b^2{\sin }^{2}x}$	Solved (Evaluated to $\frac{\pi }{2ab}$)
${\int }_0^{\pi /2}\log (\sin x)dx$	Solved (Full derivation)
${\int }_0^{\pi /4}\frac{\log (1+\tan x)}{1+x^2}dx$	Solved (Full derivation)
${\int }_0^{\pi }x{\cos }^{4}xdx$	Solved (Setup using properties of definite integral)

B. Unsolved Practice Problems (13 problems)

Problem
${\int }_0^{\pi /2}\frac{dx}{4+5\sin x}$
${\int }_0^{\pi /2}\frac{dx}{5+4\sin x}$
${\int }_0^{\pi /2}\frac{dx}{3+5\cos x}$
${\int }_0^{\pi /2}\frac{\cos xdx}{(1+\sin x)(2+\sin x)}$
${\int }_0^{\pi /2}\frac{x\sin x\cos xdx}{(a^2{\cos }^{2}x+b^2{\sin }^{2}x{)}^2}$
${\int }_{\alpha }^{\beta }\frac{dx}{(x-\alpha )\sqrt{x-2x-x^2}}$
${\int }_0^1\frac{dx}{(1+x)\sqrt{1+2x-x^2}}$
${\int }_0^{\pi /2}\log \sin (\frac{\pi }{2}x)dx$
${\int }_0^{\pi }x\log \sin xdx$
${\int }_0^1\frac{\log x}{\sqrt{1-x^2}}dx$
${\int }_0^{\pi /2}{\cos }^{6}xdx$ (Wallis’s Theorem application)
${\int }_0^{\pi /2}{\sin }^{8}xdx$ (Wallis’s Theorem application)
${\int }_0^{\pi }x{\sin }^{4}xdx$

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III. Reduction Formulas (Total: 8 problems)

A. Solved / Derived Formulas and Evaluations (4 problems)

Problem
Find the reduction formula for $\int {\sin }^{n}xdx$
Find the reduction formula for $\int {\tan }^{n}xdx$
Find the value of ${\int }_0^{\pi /4}{\tan }^{6}xdx$
Find the reduction formula for $\int e^{ax}{\cos }^{n}xdx$

B. Unsolved Derivation Tasks (4 problems)

Problem
Find the reduction formula for $\int {\cos }^{n}xdx$
Find the reduction formula for $\int {\sec }^{n}xdx$
Find the reduction formula for $\int {\cot }^{n}xdx$
Obtain the reduction formula for $\int {\sin }^{m}x{\cos }^{n}xdx$

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IV. Beta and Gamma Functions (Total: 12 problems)

A. Solved / Proved Relations and Evaluations (4 problems)

Problem	Status
Show that ${\int }_0^{\pi /2}{\sin }^{2m-1}\theta {\cos }^{2n-1}\theta d\theta =\frac{\Gamma (m)\Gamma (n)}{2\Gamma (m+n)}$	Proved
Evaluate ${\int }_0^{\pi /2}{\sin }^3\theta {\cos }^3\theta d\theta$	Solved (Evaluated to $\frac{1}{45}$)
Evaluate ${\int }_0^{\infty }{e}^{-x^2}dx$	Solved (Evaluated to $\frac{\sqrt{\pi }}{2}$)
Evaluate ${\int }_0^2\sqrt{x(2-x)}dx$	Solved (Evaluated to $\frac{\pi }{2}$)

B. Unsolved Evaluation Tasks (8 problems)

Problem
${\int }_0^{\infty }{e}^{-x^n}dx$
${\int }_0^{\pi /2}\sqrt{\tan \theta }d\theta$
${\int }_0^{\pi /2}{\sin }^6\theta {\cos }^6\theta d\theta$
${\int }_0^1{x}^3(1-x{)}^{4}dx$
${\int }_0^1{x}^2\sqrt{1-x}dx$
${\int }_0^{2}x(2-x{)}^{1/4}dx$
${\int }_0^1\frac{dx}{\sqrt{1-x^4}}$
${\int }_0^1\frac{dx}{\sqrt{1-x^n}}$

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V. Applications of Integration (Total: 12 problems)

All application problems have the integral setup and initial steps provided in the sources, though final numerical integration may be required to complete them.

Problem (Area, Length, Volume, Surface Area)
Area: Find the area enclosed by the curve $a^2{x}^2=y^3(2a-y)$
Area: Find the area bounded by $y^2-4x-4=0$ and $y^2+4x-4=0$
Area: Find the area bounded by the astroid $x^{2/3}+y^{2/3}=a^{2/3}$
Length: Find the perimeter of the loop of $3a{y}^2=x(x-a{)}^2$
Length: Find the entire length of the cardioid $r=a(1+\cos \theta )$
Length: Find the length of the curve $x=a{\cos }^{3}t,y=a{\sin }^{3}t$ (Astroid)
Volume (X-axis): Find the volume formed by the revolution of the loop of $y^2(a+x)=x^2(3a-x)$ about the x-axis
Volume (Y-axis): Find the volume generated by revolving the parabola $y^2=4ax$ along the y-axis
Volume (X-axis): Find the volume generated by revolving the parabola $y^2=4ax$ along the x-axis
Volume (Polar): Find the volume generated by the revolution of the cardioid $r=a(1+\cos \theta )$ about the initial line
Surface Area (X-axis): Find the surface area generated by revolving the parabola $y^2=4ax$ about the x-axis
Surface Area (Polar): Find the surface area generated by the revolution of the cardioid $r=a(1+\cos \theta )$ about the initial line