Integration - Trigonometric - Several Examples

Integration with trigonometric substitutions - Several Examples

Integration By Parts - Several Examples.

THE FORMULA FOR INTEGRATION BY PARTS

Let $u(x), v(x)$ be two functions of $x$.
$$ \int u(x) v'(x)\,dx=u(x) v(x)-\int u'(x) v(x)\,dx$$
or the 'quick formula' 
$$ \int uv'=uv-\int u'v$$

[Remark: This is like the product rule for derivatives]

Prove the series result - exponential series $$\left( \sum_{k=0}^\infty {u^k\over k!} \right) \left( \sum_{l=0}^\infty {v^l\over l!} \right) = \sum_{m=0}^\infty {(u+v)^m\over m!} $$ where $u,v\in C$.

Prove the series result - exponential series

$$\left( \sum_{k=0}^\infty {u^k\over k!} \right) \left( \sum_{l=0}^\infty {v^l\over l!} \right)
= \sum_{m=0}^\infty {(u+v)^m\over m!} $$

where $u,v\in C$.

College Algebra : Logarithms and Exponents manipulation and Graphs questions

QUESTIONS:

Solve each logarithmic equation.
Q14. $\displaystyle x=\log_8 \sqrt[4]{8}$
Q20. $\displaystyle x=12^{\log_{12}5}$
Q24. $\displaystyle \log_x {1 \over 16}=-2$

Sketch the graph of $f(x)=\log_2 x$. Then refer to it to graph these functions.

Q34. $\displaystyle f(x)=\log_2(x+3)$
Q42. $\displaystyle f(x)=\log_2{x\over 2}$

Q50. Sketch the graph $\displaystyle f(x)=\log_2x^2$

Use the properties of logarithms to rewrite each expression. Simplify the result if possible. Assume all variables represent positive real numbers.

Q58. $\displaystyle  \log_3{4p\over q}$
Q61. $\displaystyle  \log_4{2x+5y}$
Q64. $\displaystyle  \log_p\sqrt[3]{{m^5 n^4\over t^2}}$

Write each expression as a single logarithm with coefficient $1$. Assume all variables represent positive real numbers.

Q66. $\displaystyle  (\log_bk-\log_bm)-\log_ba$
Q68. $\displaystyle  {1\over 2} \log_yp^3q^4 -{2\over 3} \log_y p^4q^3$
Q70. $\displaystyle  \log_b(2y+5)-{1\over 2} \log_b(y+3)$

SOLUTIONS ==================

Complex functions & Limits: (2) Let $a,b$ be complex constants. Show that $$\lim_{z\rightarrow z_0} (az+b)=az_0+b$$ (7) Use the definition of limit to prove that if $\displaystyle \lim_{z\rightarrow z_0} f(z)=w_0$ then $$\displaystyle \lim_{z\rightarrow z_0} \left|f(z)\right|=\left|w_0\right|$$

QUESTIONS:

(2)  Let $a,b$ be complex constants. Show that $$\lim_{z\rightarrow z_0} (az+b)=az_0+b$$

(7) Use the definition of limit to prove that if $\displaystyle \lim_{z\rightarrow z_0} f(z)=w_0$ then 

$$\displaystyle \lim_{z\rightarrow z_0} \left|f(z)\right|=\left|w_0\right|$$

APPENDIX  ==============================

Complex Functions (Differentiability) Show that $f'(z)$ does not exist at any point $z$ when (a) $f(z)=\bar{z}$ (b) $f(z)=Re (z)$ (c) $f(z)=Im (z)$

QUESTION: Show that $f'(z)$ does not exist at any point $z$ when
(a)  $f(z)=\bar{z}$
(b)  $f(z)=Re (z)$
(c)  $f(z)=Im (z)$

APPENDIX: SOME ADDITIONAL SUPPORT FILES.

LU Decomposition, Matrices. Find a matrix $A$ that admits NO $LU$ decomposition, even if we only require that $L$ is lower triangular (not necessarily unit) and $U$ is upper triangular. Justify your answer.

QUESTION: Find a matrix $A$ that admits NO $LU$ decomposition, even if we only require that $L$ is lower triangular (not necessarily unit) and $U$ is upper triangular. Justify your answer.

Integration using substitution. (5) $\displaystyle\int_{-1}^1 {e^{\arctan x} \over 1+x^2} dx$ (12) $\displaystyle\int\sin x \cos(\cos x) dx$ (15) $\displaystyle\int_0^{1/2} {x\over \sqrt{1-x^2}} dx$ (18) $\displaystyle\int {e^{2x}\over 1+e^{4x}} dx$ (19) $\displaystyle\int e^{x+e^x} dx$ (26) $\displaystyle\int {3x^2-2\over x^3-2x-8} dx$ (27) $\displaystyle\int \cot x \ln (\sin x) dx$ (43) $\displaystyle\int e^x \sqrt{1+e^x} dx$ (34) $\displaystyle\int_{\pi/4}^{\pi/2} {1+4\cot x\over 4-\cot x} dx$

QUESTIONS: Integration using substitutions.

(5)    $\displaystyle\int_{-1}^1 {e^{\arctan x} \over 1+x^2} dx$
(12)  $\displaystyle\int\sin x \cos(\cos x)  dx$
(15)  $\displaystyle\int_0^{1/2} {x\over \sqrt{1-x^2}} dx$
(18)  $\displaystyle\int {e^{2x}\over 1+e^{4x}} dx$
(19)  $\displaystyle\int e^{x+e^x} dx$
(26)  $\displaystyle\int {3x^2-2\over x^3-2x-8} dx$
(27)  $\displaystyle\int \cot x \ln (\sin x) dx$
(43)  $\displaystyle\int e^x \sqrt{1+e^x} dx$
(34)  $\displaystyle\int_{\pi/4}^{\pi/2} {1+4\cot x\over 4-\cot x} dx$

SOLUTIONS