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• Notes

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