MATH 260 Week 7 Lab

MATRH260—Week 7 Lab			   Name:	
Logarithmic Integrals

Examine each example below then answer questions 1-3.


 Integral: ∫▒〖1/x dx〗			 Integral:∫▒〖1/(x-3) dx〗

    Solution: ln |x| +C		Solution: ln⁡|x+3|+C

 Integral:∫▒〖2x/(x^2+4) dx〗			  Integral:∫▒〖1/(2x-3) dx〗

    Solution: ln|x^2+4|+C		    Solution: 1/2 ln|2x-3|+C


1)  What is the formula for finding  ∫▒du/u?

2)  How is substitution being used to solve examples  and ?  Show u and du.
	
3)  Why can’t we find the integral∫▒〖1/x dx〗 using the following set-up?    ∫▒〖1/x dx=∫▒〖x^(-1) dx〗〗

4)   Fill in the blanks below with the correct information for the following integral:   ∫▒〖3/(4x+5) dx〗
u =		 du =	
 Substitution =		   Integration =	
Final answer =	

5)   Fill in the blanks below with the correct information for the following integral:   ∫▒〖e^(3x^2 )/(x^3-5) dx〗
u =		du =	
 Substitution =		   Integration =	
Final answer =	

6)   Fill in the blanks below with the correct information for the following integral:   ∫▒〖(6xe^(3x^2 ))/(e^(3x^2 )-5) dx〗
u =		du =	
 Substitution =		   Integration =	
Final answer =	


Exponential Integrals

Examine each example below and answer questions 7-10.

 Integral: ∫▒〖e^x dx〗		 Integral: ∫▒〖〖2xe〗^(x^2 ) dx〗

     Solution: e^x+C		    Solution: e^(x^2 )+C

 Integral: ∫▒〖e^3x dx〗		 Integral: ∫▒〖〖sec〗^2 (2x)e^(tan⁡(2x)) dx〗

    Solution: 1/3 e^3x+C		    Solution: 1/2 e^(tan⁡(2x))+C


7)  What is the formula for finding  ∫▒〖e^u du〗?  

8)  How is substitution being used to help solve examples , , and ?  
     Show u, du, and the substituted integral for each.
		
		
		

9) Fill in the blanks below with the correct information for the following integral:   ∫▒〖5x^2 e^(x^3 ) dx〗
u =		du =	
 Substitution =	
   Integration =	
Final answer =	


10) Fill in the blanks below with the correct information:  ∫▒〖sec^2 (8x) e^(3tan(8x)) dx〗
u =		du =	
 Substitution =	
   Integration =	
Final answer =	


Bandwidth of a Series Resonance Circuit

The area under the curve shows all acceptable signals with frequency greater than f L and less than
fH  that can pass through the resonance circuit. This is used in radio receivers to tune for different channels.
 

11)  If the following equation I(f)represents the curve for a frequency bandwidth similar to the one
above, find the area under the curve from f L= .8 H to f H. = 1.2 H. Round all values to the 100ths.
■(I(f)=1/(.2√2π)   ⋅  e^((-(f-  1  )^2)/(2(.2)^2 ))@), xmin= -2,  xmax = 3,  ymin= -2,  ymax = 3

Trigonometric Integrals

Some of the trig integrals are the reverse of their derivatives.

∫▒cosxdx = sin x + C	∫▒sinx dx = -cos x + C			

∫▒〖sec^2 x〗dx = tan x + C	∫▒〖csc^2 x〗 dx = -cot x + C

∫▒secxtanxdx = sec x + C	∫▒cscx   cot  xdx = -csc x + C


And some require reciprocal identities and integrate to natural logs.

∫▒secxdx = ln |sec x + tan x| + C	∫▒cscx dx = -ln |csc x + cot x| + C

∫▒cotx dx = ln|sin x| + C		∫▒tanx dx = -ln|cos x| + C


All should be memorized;remember that you can always check an integrandby differentiating.

Sine and Cosine

 Integral: ∫▒〖sin⁡(2x)dx〗			 Integral: ∫▒〖cos⁡(4x)dx〗

    Solution: -1/2  cos⁡(2x)+C			    Solution:1/4  sin⁡(4x)+C

 Integral:∫▒〖-x sin(x^2+5)dx〗		 Integral: ∫▒〖9x^2  cos⁡(x^3-15)dx〗

    Solution: 1/2  cos⁡(x^2+5)+C		    Solution: 1/3  sin⁡(x^3-15) + C


12)  Use u-substitution and the above examples to find∫▒〖e^2x  sin⁡(e^2x )dx〗

13) Use u-substitution to find∫▒〖3x cos(x^2+8)  dx〗

Secant and Cosecant, Tangent and Cotangent

Examine each example below and answer the following questions.

	Integral	Solution
i.	∫▒〖sec⁡(6x)dx〗	1/6 ln|sec⁡(6x)+tan⁡(6x)|+C
ii.	∫▒〖x^4  csc⁡〖(x^5-7)dx 〗 〗	-1/5 ln|csc(x^5-7)+cot(x^5-7) |+C
iii	∫▒e^□(1/2 x)  sec(e^□(1/2 x) )tan(e^□(1/2 x) )dx	2sec(e^□(1/2 x) )+C
iv.	∫▒〖csc⁡(10x)  cot⁡〖(10x)   dx〗 〗	-1/10  csc⁡(10x)+C
v.	∫▒cot⁡〖(10x)   dx〗 	1/10  ln⁡|sin⁡(10x)|+C
vi.	∫▒〖2x tan〗⁡〖(10x^2 )   dx〗 	-1/10 ln⁡|cos⁡(10x^2)|+C

*Note: absolute value symbols ensure that the argument for the natural logarithm is always positive and the result is real.
14) Usedifferentiation to show that each of the solutions for the integrals i., iii., v., vi. aboveis true.

15)  If∫▒〖cot⁡(u)du=ln⁡|sin⁡u |+C〗what does∫▒〖cot⁡(1/2 x)dx〗 =

16) Graph the function sin(ex) and find the area from 0 to π/4.  
xmin = -1, x-max = 5, y min=-3, y-max = 2, radian mode.

17)∫▒〖3x^5 csc(x^6 )dx〗

18)∫▒〖e^(-3x) sec⁡(e^(-3x) )tan(e^(-3x) )dx〗

19)Break the integral into three less complicated integrals, then integrate.
        Hint: rewrite using the idea that (a+b)/c=a/c+b/c , reduce, rewrite using reciprocal identities, 
then integrate.

∫▒〖(sin^2 (x)+sin(x)+cos(x))/(sin^3 (x)) dx〗