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شائع حساب التفاضل والتكامل >

integral of sqrt((sin(x))/(cos(x)))

الحلّ

\int \frac{sin ( x )}{cos ( x )} d x

الحلّ

22 (- \frac{1}{8} (ln 2 tan (x) + 22 tan (x) + 2 - 2 arctan (2 tan (x) + 1)) + \frac{1}{8} (ln 2 tan (x) - 22 tan (x) + 2 + 2 arctan (2 tan (x) - 1))) + C

خطوات الحلّ

\int \frac{sin ( x )}{cos ( x )} d x

\frac{sin ( x )}{cos ( x )} = tan (x)

:Use the basic trigonometric identity

= \int tan (x) d x

Apply Trigonometric Substitution

= \int \frac{u}{1 + u ^{2}} d u

فعّل طريقة التكامل بالتعويض

= \int \frac{2 v}{4 + v ^{2}} d v

\int a \cdot f (x) d x = a \cdot \int f (x) d x

:قم بإخراج الثابت

= 2 \cdot \int \frac{v}{4 + v ^{2}} d v

فعّل طريقة التكامل بالتعويض

= 2 \cdot \int \frac{2 w ^{2}}{4 + w ^{4}} d w

\int a \cdot f (x) d x = a \cdot \int f (x) d x

:قم بإخراج الثابت

= 2 \cdot 2 \cdot \int \frac{w ^{2}}{4 + w ^{4}} d w

\frac{w ^{2}}{4 + w ^{4}}

خذ الكسر الجزئي لـ:

- \frac{w}{4 ( w ^{2} + 2 w + 2 )} + \frac{w}{4 ( w ^{2} - 2 w + 2 )}

= 2 \cdot 2 \cdot \int - \frac{w}{4 ( w ^{2} + 2 w + 2 )} + \frac{w}{4 ( w ^{2} - 2 w + 2 )} d w

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

:فعّل الميزة الخطيّة

= 2 \cdot 2 (- \int \frac{w}{4 ( w ^{2} + 2 w + 2 )} d w + \int \frac{w}{4 ( w ^{2} - 2 w + 2 )} d w)

\int \frac{w}{4 ( w ^{2} + 2 w + 2 )} d w = \frac{1}{8} (ln w^{2} + 2 w + 2 - 2 arctan (w + 1))

\int \frac{w}{4 ( w ^{2} - 2 w + 2 )} d w = \frac{1}{8} (ln w^{2} - 2 w + 2 + 2 arctan (w - 1))

= 2 \cdot 2 (- \frac{1}{8} (ln w^{2} + 2 w + 2 - 2 arctan (w + 1)) + \frac{1}{8} (ln w^{2} - 2 w + 2 + 2 arctan (w - 1)))

استبدل مجددًا

= 2 \cdot 2 (- \frac{1}{8} (ln (2 tan (x))^{2} + 2 2 tan (x) + 2 - 2 arctan (2 tan (x) + 1)) + \frac{1}{8} (ln (2 tan (x))^{2} - 2 2 tan (x) + 2 + 2 arctan (2 tan (x) - 1)))

2 \cdot 2 (- \frac{1}{8} (ln (2 tan (x))^{2} + 2 2 tan (x) + 2 - 2 arctan (2 tan (x) + 1)) + \frac{1}{8} (ln (2 tan (x))^{2} - 2 2 tan (x) + 2 + 2 arctan (2 tan (x) - 1)))

بسّط:

22 (- \frac{1}{8} (ln 2 tan (x) + 22 tan (x) + 2 - 2 arctan (2 tan (x) + 1)) + \frac{1}{8} (ln 2 tan (x) - 22 tan (x) + 2 + 2 arctan (2 tan (x) - 1)))

= 22 (- \frac{1}{8} (ln 2 tan (x) + 22 tan (x) + 2 - 2 arctan (2 tan (x) + 1)) + \frac{1}{8} (ln 2 tan (x) - 22 tan (x) + 2 + 2 arctan (2 tan (x) - 1)))

أضف ثابت التكامل للحل

= 22 (- \frac{1}{8} (ln 2 tan (x) + 22 tan (x) + 2 - 2 arctan (2 tan (x) + 1)) + \frac{1}{8} (ln 2 tan (x) - 22 tan (x) + 2 + 2 arctan (2 tan (x) - 1))) + C

رسم

أمثلة شائعة

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