Question and Answers Forum

All Questions      Topic List

Integration Questions

Previous in All Question      Next in All Question      

Previous in Integration      Next in Integration      

Question Number 52164 by pooja24 last updated on 04/Jan/19

find:  ∫_0 ^Π (cos^6 θ −cos^4 θ) dθ  plase help me in cinding this And also  explain if possible

find:Π0(cos6θcos4θ)dθplasehelpmeincindingthisAndalsoexplainifpossible

Commented by prakash jain last updated on 04/Jan/19

∫_0 ^(π/2) sin^m xcos^n xdx  =(([(n−1)(n−3)..1 or 2]](m−1)(m−3)..1or2])/((m+n)(m+n−2)... 1 or 2))K  whereK=(π/2) if both m,n are even               K=1 otherwise  I=∫_0 ^(π/2) cos^6 xdx−∫_0 ^(π/2) cos^4 xdx  +∫_(π/2) ^π cos^6 xdx−∫_(π/2) ^π cos^4 xdx  ∫_(π/2) ^π cos^6 xdx=∫_0 ^(π/2) sin^6 xdx=∫_0 ^(π/2) cos^6 xdx  ∫_(π/2) ^π cos^4 xdx=∫_0 ^(π/2) sin^4 xdx=∫_0 ^(π/2) cos^4 xdx  I=2[∫_0 ^(π/2) cos^6 xdx−∫_0 ^(π/2) cos^4 xdx]  =2[((5∙3∙1)/(6∙4∙2))−((3∙1)/(4∙2))](π/2)  ∵m=6 even  =2[(5/(16))−(3/8)](π/2)  =−(π/(16))

0π/2sinmxcosnxdx=[(n1)(n3)..1or2]](m1)(m3)..1or2](m+n)(m+n2)...1or2KwhereK=π2ifbothm,nareevenK=1otherwiseI=0π/2cos6xdx0π/2cos4xdx+π/2πcos6xdxπ/2πcos4xdxπ/2πcos6xdx=0π/2sin6xdx=0π/2cos6xdxπ/2πcos4xdx=0π/2sin4xdx=0π/2cos4xdxI=2[0π/2cos6xdx0π/2cos4xdx]=2[5316423142]π2m=6even=2[51638]π2=π16

Commented by maxmathsup by imad last updated on 04/Jan/19

let I =∫_0 ^π (cos^6 x−cos^4 x)dx ⇒I =∫_0 ^π (cos^4 x(1−sin^2 x−cos^4 x)dx  =−∫_0 ^π  sin^2 x cos^4 x dx =−∫_0 ^π ((1−cos(2x))/2)(((1+cos(2x))/2))^2 dx  =−(1/8) ∫_0 ^π (1−cos^2 (2x))(1+cos(2x))dx  =−(1/8) ∫_0 ^π (1−((1+cos(4x))/2))(1+cos(2x))dx  =−(1/8) ∫_0 ^π (((1−cos(4x))/2))(1+cos(2x))dx  =−(1/(16)) ∫_0 ^π (1+cos(2x)−cos(4x)−cos(2x)cos(4x))dx  =−(π/(16)) −(1/(16)) ∫_0 ^π cos(2x)dx+(1/(16)) ∫_0 ^π cos(4x)dx +(1/(16)) ∫_0 ^π cos(2x)cos(4x)dx  =−(π/(16)) −(1/(32))[sin(2x)]_0 ^π  +(1/(64))[sin(4x)]_0 ^π    +(1/(32)) ∫_0 ^π (cos(6x)+cos(2x))dx  =−(π/(16)) +0 +0 +0 =−(π/(16)) .

letI=0π(cos6xcos4x)dxI=0π(cos4x(1sin2xcos4x)dx=0πsin2xcos4xdx=0π1cos(2x)2(1+cos(2x)2)2dx=180π(1cos2(2x))(1+cos(2x))dx=180π(11+cos(4x)2)(1+cos(2x))dx=180π(1cos(4x)2)(1+cos(2x))dx=1160π(1+cos(2x)cos(4x)cos(2x)cos(4x))dx=π161160πcos(2x)dx+1160πcos(4x)dx+1160πcos(2x)cos(4x)dx=π16132[sin(2x)]0π+164[sin(4x)]0π+1320π(cos(6x)+cos(2x))dx=π16+0+0+0=π16.

Commented by maxmathsup by imad last updated on 04/Jan/19

error of typo at first line I =∫_0 ^π {cos^4 x(1−sin^2 x)−cos^4 x}dx...

erroroftypoatfirstlineI=0π{cos4x(1sin2x)cos4x}dx...

Commented by pooja24 last updated on 05/Jan/19

thank you

thankyou

Answered by tanmay.chaudhury50@gmail.com last updated on 04/Jan/19

−cos^4 θ(1−cos^2 θ)  =−cos^4 θsin^2 θ  =−(((1+cos2θ)/2))^2 (((1−cos2θ)/2))  =−(1/8)(1+cos2θ)(1−cos^2 2θ)  =−(1/8)(1+cos2θ)(1−((1+cos4θ)/2))  =((−1)/(16))(1+cos2θ)(2−1−cos4θ)  =((−1)/(16))(1+cos2θ)(1−cos4θ)  =((−1)/(16))(1−cos4θ+cos2θ−cos2θcos4θ)  =((−1)/(16))(1−cos4θ+cos2θ−((cos6θ+cos2θ)/2))  =((−1)/(32))(2−2cos4θ+2cos2θ−cos6θ−cos2θ)  =(1/(32))(cos6θ+2cos4θ−cos2θ−2)  now   (1/(32))∫_0 ^π cos6θ+2cos4θ−cos2θ−(1/(16))∫_0 ^π dθ  (1/(32))∣((sin6θ)/6)+((2sin4θ)/4)−((sin2θ)/2)∣_0 ^π −(1/(16))×π                              _  ⇓_(intregal value zero_ )     −(1/(16))×π    answer is =((−π)/(16))                       [sinnπ=0]  pls chek...

cos4θ(1cos2θ)=cos4θsin2θ=(1+cos2θ2)2(1cos2θ2)=18(1+cos2θ)(1cos22θ)=18(1+cos2θ)(11+cos4θ2)=116(1+cos2θ)(21cos4θ)=116(1+cos2θ)(1cos4θ)=116(1cos4θ+cos2θcos2θcos4θ)=116(1cos4θ+cos2θcos6θ+cos2θ2)=132(22cos4θ+2cos2θcos6θcos2θ)=132(cos6θ+2cos4θcos2θ2)now1320πcos6θ+2cos4θcos2θ1160πdθ132sin6θ6+2sin4θ4sin2θ20π116×πintregalvaluezero116×πansweris=π16[sinnπ=0]plschek...

Answered by tanmay.chaudhury50@gmail.com last updated on 04/Jan/19

Terms of Service

Privacy Policy

Contact: info@tinkutara.com