1-let-f-a-a-tanx-dx-with-a-gt-0-find-a-explicit-form-for-f-x-2-find-also-g-a-dx-a-tanx-3-calculate-2-tanx-dx-and-dx-2-tanx- Tinku Tara June 3, 2023 Integration 0 Comments FacebookTweetPin Question Number 69044 by mathmax by abdo last updated on 18/Sep/19 1)letf(a)=∫a+tanxdxwitha>0findaexplicitformforf(x)2)findalsog(a)=∫dxa+tanx3)calculate∫2+tanxdxand∫dx2+tanx Answered by mind is power last updated on 18/Sep/19 u=a+tgxx=arctg(u2−a)dx=2udu1+(u2−a)2∫a+tan(x)dx=∫2u2du(u2−a)2+1=∫2u2duu4−2au2+a2+1=∫2u2du(u2+a2+1)2−(2a+2a2+1)u2=∫2u2du(u2−(2a+2a2+1)u+a2+1)(u2+2a+a2+1u+a2+1)β=2a+2a2+1η=a2+1∫2u2(u2−βx+η)(x2+βx+η)=ax+bx2−βx+η+cx+dx2+βx+ηa+c=0b+d=0β(a−c)+d+b=2η(c+a)+β(−d+b)=0⇒a=−cb=−d,−2cβ=2b=d,c=−1β,a=1β,b=d=0∫xβ(x2−βx+η)dx−∫dxβ(x2+βx+η)=1β∫dx(x−β2)2+η−β24−1β∫dx(x+β2)2+η−β24=1βη−β24tan−1(x−β2η−β24)−1βη−β24tan−1(x+β2η−β24)+c∫a+tan(x)dx=1βη−β24[tan−1(a+tg(x)−β2η−β24)−tan−1(a+tg(x)+β2η−β24)+cf(a)=12a+2a2+1a2+1−a2[tan−1(a+tan(x)−2a+2a2+12a2+1−a2)−tan−1(a+tan(x)+2a+2a2+12a2+1−a2)]f′(a)=∫dda(a+tan(x)dx)=12∫dxa+tan(x)⇒∫dxa+tan(x)=2f′(a)″verrylongexpression″3)∫2+tan(x)dx=f(2)∫dx2+tg(x)=2f′(2) Commented by turbo msup by abdo last updated on 19/Sep/19 thankyousir. Commented by mind is power last updated on 19/Sep/19 y′rewelcom Terms of Service Privacy Policy Contact: info@tinkutara.com FacebookTweetPin Post navigation Previous Previous post: Question-69034Next Next post: Question-134583 Leave a Reply Cancel replyYour email address will not be published.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.