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 205725 by NasaSara last updated on 28/Mar/24

Answered by Berbere last updated on 29/Mar/24

∫_0 ^2 ∫_0 ^3 [xy]dydx=^(xy=t) ∫_0 ^2 ∫_0 ^(3x) [t](dt/x)dx  =∫_0 ^2 (1/x)∫_0 ^(3x) [t]dtdx=  =Σ_(k=1) ^5 {∫_(k/3) ^((k+1)/3) (1/x){Σ_(j=1) ^(k−1) ∫_j ^(j+1) [t]+∫_k ^(3x) [t]}dt=  isolat k=1 ;and k≥2  ∫_(1/3) ^(2/3) (1/x){3x−1}=1−[ln(2))]  .Σ_(k=2) ^5 ∫_(k/3) ^((k+1)/2) (1/x){Σ_(j=1) ^(k−1) j+∫_k ^(3x) kdt}dx+1−ln(2)=  Σ_(k=2) ^5 ∫_(k/3) ^((k+1)/3) (1/x){(((k−1)k)/2)+k(3x−k)}dx  =Σ_(k=2) ^5 k+Σ_(k=2) ^5 ∫_(k/3) ^((k+1)/3) (1/x)(−(((1+k)k)/2))dx+1−ln(2)  =15−Σ_(k=2) ^5 (((1+k)k)/2)ln(((k+1)/k))−ln(2)  =15−3ln((3/2))−6ln((4/3))−10ln((5/4))−15ln((6/5))−ln(2)  =15+3ln(3)+2ln(2)+4ln(4)+5ln(5)−15ln(6)  =15−ln((6^(15) /(3^3 .2^2 .4^4 .5^5 )))=15−ln(((3^(12) .2^5 )/(.5^5 )))=15−ln(((17006112)/(3125)))

0203[xy]dydx=xy=t0203x[t]dtxdx=021x03x[t]dtdx==5k=1{k3k+131x{k1j=1jj+1[t]+k3x[t]}dt=isolatk=1;andk213231x{3x1}=1[ln(2))].5k=2k3k+121x{k1j=1j+k3xkdt}dx+1ln(2)=5k=2k3k+131x{(k1)k2+k(3xk)}dx=5k=2k+5k=2k3k+131x((1+k)k2)dx+1ln(2)=155k=2(1+k)k2ln(k+1k)ln(2)=153ln(32)6ln(43)10ln(54)15ln(65)ln(2)=15+3ln(3)+2ln(2)+4ln(4)+5ln(5)15ln(6)=15ln(61533.22.44.55)=15ln(312.25.55)=15ln(170061123125)

Commented by NasaSara last updated on 04/Apr/24

thank you so much

Terms of Service

Privacy Policy

Contact: info@tinkutara.com