calculate-0-pi-cos-t-1-2-cost-2-dt- Tinku Tara June 4, 2023 Integration 0 Comments FacebookTweetPin Question Number 36946 by maxmathsup by imad last updated on 07/Jun/18 calculateφ(λ)=∫0πcos(t)1−2λcost+λ2dt Commented by math khazana by abdo last updated on 08/Jun/18 changementtan(t2)=xgiveφ(λ)=∫0∞1−x21+x21−2λ1−x21+x22dx1+x2=2∫0∞1−x2(1+x2){1+x2−2λ+2λx2}dx=2∫0∞1−x2(1+x2)((2λ+1)x2+1−2λ)dxletdecomposeF(x)=1−x2(1+x2){(2λ+1)x2+1−2λ)F(x)=ax+bx2+1+cx+d(2λ+1)x2+1−2λF(−x)=F(x)⇒−ax+bx2+1+−cx+d(2λ+1)x2+1−2λ=F(x)⇒a=c=0⇒F(x)=bx2+1+d(2λ+1)x2+1−2λlimx→+∞x2F(x)=−12λ+1=b+d2λ+1⇒−1=(2λ+1)b+d⇒d=−1−(2λ+1)bF(x)=bx2+1−1+(2λ+1)b(2λ+1)x2+1−2λF(o)=11−2λ=b−1+(2λ+1)b1−2λ⇒1=(1−2λ)b−1−(2λ+1)b=−4λb−1⇒2=−4λb⇒b=−12λ(λ≠0andλ≠12)∫0+∞F(x)dx=−12λ∫0∞{1x2+1−1−2λ+12λ(2λ+1)x2+1−2λ}dx=−12λ∫0∞dx1+x2−12λ.−12λ∫0∞dx(2λ+1)x2+1−2λ=−12λπ2+14λ2∫0∞dx(2λ+1)x2+1−2λbut∫0∞dx(2λ+1)x2+1−2λ=12λ+1∫0∞dxx2+1−2λ1+2λif1−2λ1+2λ>0weusethechang.x=1−2λ1+2λu=12λ+1∫0∞11−2λ1+2λ(1+x2)1−2λ1+2λdu=12λ+11+2λ1−2λ1−2λ1+2λπ2=π2(2λ+1)1+2λ1−2λ=π211−4λ2⇒∫0∞F(x)dx=−π4λ+14λ2π211−4λ2=−π4λ+π8λ21−4λ2φ(λ)=2∫0∞F(x)dx⇒φ(λ)=π4λ21−4λ2−π2λ. Commented by math khazana by abdo last updated on 08/Jun/18 if1−2λ1+2λ<0weget∫0∞dxx2+1−2λ1+2λ=∫0∞dxx2−(−1−2λ1+2λ)2andweusethedecompositon… Terms of Service Privacy Policy Contact: info@tinkutara.com FacebookTweetPin Post navigation Previous Previous post: n-positive-integer-when-dividen-by-7-give-remainder-4-and-when-divided-by-4-give-remainder-2-find-the-value-of-n-Next Next post: calculate-D-xy-dxdy-with-D-x-y-R-2-x-y-2-2x-and-xy-0- Leave a Reply Cancel replyYour email address will not be published. Required fields are marked *Comment * Name * Save my name, email, and website in this browser for the next time I comment.