expand sin(e^x+sqrt(x))(1+cos(x))^{14}

الحلّ

expand

sin (e^{x} + x) (1 + cos (x))^{14}

sin (e^{x} + x) + 14 sin (e^{x} + x) cos (x) + 91 cos^{2} (x) sin (e^{x} + x) + 364 cos^{3} (x) sin (e^{x} + x) + 1001 cos^{4} (x) sin (e^{x} + x) + 2002 cos^{5} (x) sin (e^{x} + x) + 3003 cos^{6} (x) sin (e^{x} + x) + 3432 cos^{7} (x) sin (e^{x} + x) + 3003 cos^{8} (x) sin (e^{x} + x) + 2002 cos^{9} (x) sin (e^{x} + x) + 1001 cos^{10} (x) sin (e^{x} + x) + 364 cos^{11} (x) sin (e^{x} + x) + 91 cos^{12} (x) sin (e^{x} + x) + 14 cos^{13} (x) sin (e^{x} + x) + cos^{14} (x) sin (e^{x} + x)

خطوات الحلّ

sin (e^{x} + x) (1 + cos (x))^{14}

(1 + cos (x))^{14} = 1 + 14 cos (x) + 91 cos^{2} (x) + 364 cos^{3} (x) + 1001 cos^{4} (x) + 2002 cos^{5} (x) + 3003 cos^{6} (x) + 3432 cos^{7} (x) + 3003 cos^{8} (x) + 2002 cos^{9} (x) + 1001 cos^{10} (x) + 364 cos^{11} (x) + 91 cos^{12} (x) + 14 cos^{13} (x) + cos^{14} (x)

= sin (x + e^{x}) (cos^{14} (x) + 14 cos^{13} (x) + 91 cos^{12} (x) + 364 cos^{11} (x) + 1001 cos^{10} (x) + 2002 cos^{9} (x) + 3003 cos^{8} (x) + 3432 cos^{7} (x) + 3003 cos^{6} (x) + 2002 cos^{5} (x) + 1001 cos^{4} (x) + 364 cos^{3} (x) + 91 cos^{2} (x) + 14 cos (x) + 1)

فعّل قانون ضرب الأقواس

= sin (e^{x} + x) \cdot 1 + sin (e^{x} + x) \cdot 14 cos (x) + sin (e^{x} + x) \cdot 91 cos^{2} (x) + sin (e^{x} + x) \cdot 364 cos^{3} (x) + sin (e^{x} + x) \cdot 1001 cos^{4} (x) + sin (e^{x} + x) \cdot 2002 cos^{5} (x) + sin (e^{x} + x) \cdot 3003 cos^{6} (x) + sin (e^{x} + x) \cdot 3432 cos^{7} (x) + sin (e^{x} + x) \cdot 3003 cos^{8} (x) + sin (e^{x} + x) \cdot 2002 cos^{9} (x) + sin (e^{x} + x) \cdot 1001 cos^{10} (x) + sin (e^{x} + x) \cdot 364 cos^{11} (x) + sin (e^{x} + x) \cdot 91 cos^{12} (x) + sin (e^{x} + x) \cdot 14 cos^{13} (x) + sin (e^{x} + x) cos^{14} (x)

= 1 \cdot sin (e^{x} + x) + 14 sin (e^{x} + x) cos (x) + 91 cos^{2} (x) sin (e^{x} + x) + 364 cos^{3} (x) sin (e^{x} + x) + 1001 cos^{4} (x) sin (e^{x} + x) + 2002 cos^{5} (x) sin (e^{x} + x) + 3003 cos^{6} (x) sin (e^{x} + x) + 3432 cos^{7} (x) sin (e^{x} + x) + 3003 cos^{8} (x) sin (e^{x} + x) + 2002 cos^{9} (x) sin (e^{x} + x) + 1001 cos^{10} (x) sin (e^{x} + x) + 364 cos^{11} (x) sin (e^{x} + x) + 91 cos^{12} (x) sin (e^{x} + x) + 14 cos^{13} (x) sin (e^{x} + x) + cos^{14} (x) sin (e^{x} + x)

1 \cdot sin (e^{x} + x) = sin (e^{x} + x) :

اضرب

= sin (e^{x} + x) + 14 sin (e^{x} + x) cos (x) + 91 cos^{2} (x) sin (e^{x} + x) + 364 cos^{3} (x) sin (e^{x} + x) + 1001 cos^{4} (x) sin (e^{x} + x) + 2002 cos^{5} (x) sin (e^{x} + x) + 3003 cos^{6} (x) sin (e^{x} + x) + 3432 cos^{7} (x) sin (e^{x} + x) + 3003 cos^{8} (x) sin (e^{x} + x) + 2002 cos^{9} (x) sin (e^{x} + x) + 1001 cos^{10} (x) sin (e^{x} + x) + 364 cos^{11} (x) sin (e^{x} + x) + 91 cos^{12} (x) sin (e^{x} + x) + 14 cos^{13} (x) sin (e^{x} + x) + cos^{14} (x) sin (e^{x} + x)

e x p an d cos ((2 n - 1) π)

e x p an d \frac{( x + 1 ) ( x + 5 )}{9 x - 9}

e x p an d π (1 - X^{2})^{2}

e x p an d (x + 1 - x^{2})^{5}

e x p an d 1 n (x^{\frac{2}{5}} y^{\frac{4}{3}})