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التكاملات التى تكون على الشكل

\(\sqrt{a^{2}-x^{2}}, \sqrt{x^{2}-a^{2}}, \sqrt{a^{2}+x^{2}}\)

 

هنستخدم الـ (Trigonometric Substitution) لكى يصبح  التكامل على صورة الـ (Trigonometric integrals)

Identity

Substitution

Expression

\(1-\sin ^{2} \theta=\cos ^{2} \theta\)

\(1+\tan ^{2} \theta=\sec ^{2} \theta\)

\(\sec ^{2} \theta-1=\tan ^{2} \theta\)

\(x=a \sin \theta, \quad-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\)\(x=a \tan \theta, \quad-\frac{\pi}{2}<\theta<\frac{\pi}{2}\)

\(x=a \sec \theta, \quad 0 \leq \theta<\frac{\pi}{2}\) or \(\pi \leq \theta<\frac{3 \pi}{2}\)

\(\sqrt{a^{2}-x^{2}}\)

\(\sqrt{a^{2}+x^{2}}\)

\(\sqrt{x^{2}-a^{2}}\)

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