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الـ (Trigonometric integrals) هى تكاملات يوجد بها (trigonometric function)

الحالة الاولى : التكامل على الشكل

Strategy for Evaluating\(\int \sin ^{m} x \cos ^{n} x d x\)

 

(a) If the power of cosine is odd (n-2 k+1) , save one cosine factor and use

\(cos ^{2} x=1-\sin ^{2} x\) to express the remaining factors in terms of sin:

\(\begin{aligned} \int \sin ^{m} x \cos ^{2 k+1} x d x &=\int \sin ^{m} x\left(\cos ^{2} x\right)^{k} \cos x d x \\ &=\int \sin ^{m} x\left(1-\sin ^{2} x\right)^{k} \cos x d x \end{aligned}\)

Then substitute  \(u = \sin x\)

(b) If the power of sine is odd (m-2 k+1) , save one sine factor and use \(\sin ^{2} x=1-\cos ^{2} x\)  to express the remaining factors in terms of cosine:

\( \int \sin ^{2 k+1} x \cos ^{n} x d x\)  and \(=\int\left(\sin ^{2} x\right)^{k} \cos ^{n} x \sin x d x\)   and \(=\int\left(1-\cos ^{2} x\right)^{k} \cos ^{n} x \sin x d x\) 

Then substitute \(u=\cos x .\) [Note that if the powers of both sinc and cosine

are odd, cither (a) or (b) can be used]

(c) If the powers of both sine and cosine are even, use the half-angle identities \( {\sin ^{2} x=\frac{1}{2}(1-\cos 2 x)}\) and \({\cos ^{2} x=\frac{1}{2}(1+\cos 2 x)}\) It is sometimes helpful to use the identity \( {\sin x \cos x=\frac{1}{2} \sin 2 x}\)



الحالة الثانية : التكامل على الشكل

 

 Strategy for Evaluating \(\int \tan ^{\pi} x \sec ^{a} x d x\) 

 (a) If the power of secant is even \((n=2 k, k \geq 2)\) , save a factor of  \(\sec ^{2} x\) and

use \(sec^{2} x=1+\tan ^{2} x\) to express the rema

ining factors in terms of \(tan x :\)

\(\int \tan ^{m} x \sec ^{2 k} x d x\) and \(=\int \tan ^{m} x(\sec ^{2} x)^{k-1} \sec ^{2} x d x\) and \(=\int \tan ^{\pi} x(1+\tan ^{2} x)^{k-1} \sec ^{2} x d x\)(b) If the power of tangent is odd \((m=2 k+1),\) save a factor of \(sec x\) \(tan x \)

and use \(\tan ^{2} x=\sec ^{2} x-1\) to express the remaining factors in terms of

\(sec x :\)

\(\int \tan ^{2 k+1} x \sec ^{n} x d x=\int\left(\tan ^{2} x\right)^{k} \sec ^{n-1} x \sec x \tan x d x\)

\(\quad=\int\left(\sec ^{2} x-1\right)^{k} \sec ^{n-1} x \sec x \tan x d x\)

Then substitute \(u=\sec x .\)

 

الحالة الثالثة:

To evaluate the integrals \((a) \int \sin m x \cos n x d x,(b) \int \sin m x \sin n x d x,\) or

(c) \(\int \cos m x \cos n x d x,\) use the corresponding identity:

(a) \( \sin A \cos B \)  and \(=\frac{1}{2}[\sin (A-B)+\sin (A+B)]\)

(b) \(\sin A \sin B\) and \(=\frac{1}{2}[\cos (A-B)-\cos (A+B)]\)

(c) \(\cos A \cos B\) and \(=\frac{1}{2}[\cos (A-B)+\cos (A+B)]\)

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