Question #346253

explain clearly.



∫(7𝑐𝑠𝑐2𝑥 + 2 sec 𝑥 tan 𝑥)𝑑𝑥

1
Expert's answer
2022-05-31T09:58:45-0400
(7csc2x+2secxtanx)dx\int(7\csc2x+2 \sec x \tan x)dx

=(72sinxcosx+2sinxcos2x)dx=\int(\dfrac{7}{2\sin x\cos x}+\dfrac{2\sin x}{\cos^2x})dx

=(7sinx2sin2xcosx+2sinxcos2x)dx=\int(\dfrac{7\sin x}{2\sin^2 x\cos x}+\dfrac{2\sin x}{\cos^2x})dx

Use uu -substitution


u=cosx,du=sinxdxu=\cos x, du=-\sin x dx

(7sinx2sin2xcosx+2sinxcos2x)dx\int(\dfrac{7\sin x}{2\sin^2 x\cos x}+\dfrac{2\sin x}{\cos^2x})dx

=(72u(1u2)2u2)du=\int(-\dfrac{7}{2u(1-u^2)}-\dfrac{2}{u^2})du

72u(1u2)=Au+B1u+C1+u-\dfrac{7}{2u(1-u^2)}=\dfrac{A}{u}+\dfrac{B}{1-u}+\dfrac{C}{1+u}

=A(1u2)+Bu(1+u)+Cu(1u)u(1u2)=\dfrac{A(1-u^2)+Bu(1+u)+Cu(1-u)}{u(1-u^2)}


72=A(1u2)+Bu(1+u)+Cu(1u)-\dfrac{7}{2}=A(1-u^2)+Bu(1+u)+Cu(1-u)


u=0:A=72u=0:A=-\dfrac{7}{2}

u=1:C=74u=-1:C=\dfrac{7}{4}

u=1:B=74u=1:B=-\dfrac{7}{4}

(72u(1u2))du\int(-\dfrac{7}{2u(1-u^2)})du

=72duu74du1u+74du1+u=-\dfrac{7}{2}\int\dfrac{du}{u}-\dfrac{7}{4}\int\dfrac{du}{1-u}+\dfrac{7}{4}\int\dfrac{du}{1+u}

=72lnu+74ln1u+74ln1+u+C1=-\dfrac{7}{2}\ln |u|+\dfrac{7}{4}\ln|1-u|+\dfrac{7}{4}\ln|1+u|+C_1

(2u2)du=2u+C2\int(-\dfrac{2}{u^2})du=\dfrac{2}{u}+C_2

Then


(72u(1u2)2u2)du\int(-\dfrac{7}{2u(1-u^2)}-\dfrac{2}{u^2})du

=72lnu+74ln1u+74ln1+u+2u+C=-\dfrac{7}{2}\ln |u|+\dfrac{7}{4}\ln|1-u|+\dfrac{7}{4}\ln|1+u|+\dfrac{2}{u}+C

So


(7csc2x+2secxtanx)dx\int(7\csc2x+2 \sec x \tan x)dx

=72lncosx+74ln1cosx+74ln1+cosx=-\dfrac{7}{2}\ln |\cos x|+\dfrac{7}{4}\ln|1-\cos x|+\dfrac{7}{4}\ln|1+\cos x|

+2cosx+C+\dfrac{2}{\cos x}+C

=72lncosx+74ln1cos2x+2cosx+C=-\dfrac{7}{2}\ln |\cos x|+\dfrac{7}{4}\ln|1-\cos^2 x|+\dfrac{2}{\cos x}+C

=72lncosx+72lnsinx+2cosx+C=-\dfrac{7}{2}\ln |\cos x|+\dfrac{7}{2}\ln |\sin x|+\dfrac{2}{\cos x}+C


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