Question

求教几道高数题求(3x^3+x)/(1+x^4)(cosx)^4(sinx*cosx)^2(cosx)^3/(sinx)

Answer 1

inv(f(x),x)表示f(x)对x积分

inv((3x^3+x)/(1+x^4),x)

=inv(1/2*(3x^2+1)/(1+x^4),x^2) t=x^2

=0.5*inv((3t+1)/(1+t^2),t)

=3/4*ln(1+t^2)+arctan(t)+C

=3/4*ln(1+x^4)+arctan(x^2)+C

(cosx)^4=cos(4x)/8+cos(2x)/2+3/8

inv((cosx)^4,x)

=sin(4x)/32+sin(2x)/4+3x/8+C

(sin(x)*cos(x))^2=(sin(2x))^2/4=(cos(4x)-1)/8

inv((sin(x)*cos(x))^2,x)

=sin(4x)/32-x/8+C

inv((cosx)^3/(sinx)^2,x)

=inv((cosx)^2/(sinx)^2,sinx) t=sin(x)

=inv((1-t^2)/t^2,t)

=-1/t-t+C

=-1/sin(x)-sin(x)+C