Expert: NUR ARINA BASILAH BINTI KAMISAN - 8/30/2008

Question
QUESTION: pleez help me to integrate sin x/cos squared x+cos x-2

ANSWER: hi taahira,

i'm a little bit confuse with your question with cos x-2.is it cos(x-2) or cos(x)-2.i assume it was cos(x)-2.

so when you have to integrate sin x/cos^2(x)+cos x-2,first step is:

let u=cos x then du=-sin xdx
now substitute it into the original equation and you will have:
-du/(u^2+u-2) or
-[1/u^2+1/u-1/2]du
and integrate this you will have:
-[-1/u+ln(u)-u/2]+c
1/u-ln(u)+u/2+c
substitute back u=cos x
1/cos x-ln(cos x)+cos x/2+c

for further information on integration of trigonometry,you can refer to this website:
http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/TrigInt.htm

good luck!!!

---------- FOLLOW-UP ----------

QUESTION: TNX FOR DA RESPONSE..SORI FOR THE AMBIGUITY..JUST ONE MORE Q...WHEN DO YOU USE LN X + C AND WHEN DO YOU USE 1/X +C...WHEN YOYU INTERGRATE THAT IS

ANSWER: dear taahira,
the rule of integration for simple equation is when you have x^a, the integration will become (x^a+1)/a+1 + c

so if you have 1/(x^2),it was also equal to x^-2.integrate this equation,you will have [x^(-2+1)]/(-2+1) + c =x^-1/-1 + c=-1/x + c

and if you have 1/x=x^-1,if you integrate it by using the formula above you will have, x^(-1+1)=x^0.this is incorrect.so for 1/x,the integration will become ln(x) + c.

try to find it in calculus note.good luck!!!

---------- FOLLOW-UP ----------

QUESTION: THANK YOU VERY MUCH FOR ALL THE RESPONSES-EXTREMELY HELPFUL!JUST 2 MORE QUESTIONS PLEASE:1-HOW DO YOU GO ABOUT INTERGRATING 1-(TAN SQUARED X)    2-FIND THE VOLUME OF THE SOLID GENERATED BY THE CURVE y=sec x and the lines y=0;x=-pi/4;x=pi/4...much appreciated!

Answer
hi taahira,
sori for the late reply.

when you integrate tan^2(x), first you have to change it to identity.as we know tan^2(x)=sec^2(x)-1 (from the trigonometry identity).so now the integration of tan^2(x) will become integration of sec^2(x)-1 and the answer will be tan(x)-x+c

for your secod queston,first you have to know how is the graph.you can clck this link to see the graph:
http://www.intmath.com/Trigonometric-graphs/4_Graphs-tangent-cotangent-secant-co
now integrate y=sec(x) from -pi/4 to pi/4 at y=0 and for volume we have a certain formla of ntegratio that you must follow.check ths wbste for more information http://www.intmath.com/Applications-integration/4_Volume-solid-revolution.php

volume=pi*integration of [sec(x)]^2 from -pi/4 to pi/4
=pi*tan(x) from -pi/4 to pi/4
=pi*[tan(pi/4)-tan(-pi/4)]
=pi*[1-(-1)]
=2pi

try to understand the integration of a solid volume more ya?good luck!!!!