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العربية

integral of (sin^2(pix))/(cos^6(pix))

الحلّ

\int \frac{sin ^{2} ( π x )}{cos ^{6} ( π x )} d x

\frac{1}{π} (\frac{tan ^{3} ( π x )}{3} + \frac{tan ^{5} ( π x )}{5}) + C

خطوات الحلّ

\int \frac{sin ^{2} ( π x )}{cos ^{6} ( π x )} d x

فعّل طريقة التكامل بالتعويض

= \int \frac{sin ^{2} ( u )}{π cos ^{6} ( u )} d u

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

:قم بإخراج الثابت

= \frac{1}{π} \cdot \int \frac{sin ^{2} ( u )}{cos ^{6} ( u )} d u

Rewrite using trig identities

= \frac{1}{π} \cdot \int \frac{tan ^{2} ( u )}{cos ^{4} ( u )} d u

\frac{1}{cos ( x )} = sec (x)

:Use the basic trigonometric identity

= \frac{1}{π} \cdot \int sec^{4} (u) tan^{2} (u) d u

Rewrite using trig identities

= \frac{1}{π} \cdot \int (1 + tan^{2} (u)) sec^{2} (u) tan^{2} (u) d u

فعّل طريقة التكامل بالتعويض

= \frac{1}{π} \cdot \int v^{2} (1 + v^{2}) d v

v^{2} (1 + v^{2})

وسٌع:

v^{2} + v^{4}

= \frac{1}{π} \cdot \int v^{2} + v^{4} d v

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

:فعّل الميزة الخطيّة

= \frac{1}{π} (\int v^{2} d v + \int v^{4} d v)

\int v^{2} d v = \frac{v ^{3}}{3}

\int v^{4} d v = \frac{v ^{5}}{5}

= \frac{1}{π} (\frac{v ^{3}}{3} + \frac{v ^{5}}{5})

استبدل مجددًا

= \frac{1}{π} (\frac{tan ^{3} ( π x )}{3} + \frac{tan ^{5} ( π x )}{5})

أضف ثابت التكامل للحل

= \frac{1}{π} (\frac{tan ^{3} ( π x )}{3} + \frac{tan ^{5} ( π x )}{5}) + C

\int sin (y) + y d y

\int \frac{- 3}{x} - \frac{9}{x ^{2}} d x

\int \frac{2 x - 3}{x ^{2} + 2 x + 5} d x

\int - cos^{3} (t) d t

\int \frac{11 x + 8}{10 x ^{2} + 5 x} d x