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Calculus/Calculus-2 help- trigonometric integrals under principles of integral evaluation chapter

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Expert: Alon Mandes - 10/18/2008


Question
Hello, Good evening, sir. I have trouble solving this three problems.
1. Rewrite sin x + cos x in the form of A sin(x+0/) And use your results together with previous exercise to evaluate
Integration of dx/sin x + cos x    .....
The previous exercise is integration of csc x dx = -ln|csc x + cot x| + c & prove integration of csc x dx= ln|tan(x/2)|+c   ((((((0/= angle= fay,, becoz I couldn' find the symbol on keyboard)))))

Please help me sir thanks

Answer
Ok Nayan, 1st of all let's write sin(x)+cos(x) as Asin(x+Ø).
You must agree that sin(x)+cos(x)=√2[(1/√2)sin(x)+(1/√2)cos(x)].
It's the same, it's only a mathmatical trick.
Now, we also know that 1/√2 is exactly sin(45°)=cos(45°). That means:
√2[(1/√2)sin(x)+(1/√2)cos(x)]=√2[sin(x)cos(45°)+cos(x)sin(45°].
& according the the tregonometric idintity :
sin(α+β)=sin(α)cos(β)+cos(α)sin(β)
We can write : √2[sin(x)cos(45°)+cos(x)sin(45°]=√2sin(x+45°).
A=√2 & Ø=45°.

As for the 2nd part :
∫dx/(sinx+cosx)=∫dx/√2sin(x+45°)=(1/√2)∫dx/sin(x+45°)=
(1/√2)∫sec(x+45°)dx=(1/√2)Ln|sec(x+45°)+tan(x+45°)|.

Alon.

Calculus

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Alon Mandes

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Kind of questions I can answer : Limits, Derivatives, Integration, Implicit functions, continuousity, differentiation ,Extremum problems, Lagrange multipliers, Gradients, Surface integrals, Multi variables functions ,Multi variables Integrals,Complex variables ,Complex functions, Curves, Trajectory integrals & Vector analyse,Divergence,Rotor & word problems. Kind of question I can't answer : Economics,Combinatorics,infinite series & convergence ,Statistics & Probabilities .

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1. I'm a team member of mathnerds (math site for answering questions) 2. I'm a team member in the Student's Union of the Technion, helping students who have problems in mathematics. 3. 2 years of experience as a math teacher in college. 4. I give free homework help for high school students in Mathematics & Physics. 5. I teach part time in collage the subjects : "Digital Signal Processing" , "Random Signals & Noise" , "Complex Functions".

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M.A in Mathematics & Bs.c in Electronics.

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