let-f-x-1-x-a-x-b-x-c-let-a-b-c-R-a-lt-b-lt-c-D-f-x-a-b-c-prove-a-b-f-x-dx-c-f-x-dx- Tinku Tara October 19, 2024 Integration 0 Comments FacebookTweetPin Question Number 212626 by Ghisom last updated on 19/Oct/24 letf(x)=1(x−a)(x−b)(x−c)leta,b,c∈R∧a<b<c⇒D(f(x))=(a,b)∪(c,∞)prove∫baf(x)dx=∫∞cf(x)dx Answered by MrGaster last updated on 02/Nov/24 ∫abf(x)dx=∫ab1(x−a)(x−b)(x−c)dx∫c∞f(x)dx=∫c∞1(x−a)(x−b)(x−c)dxx=a+(b−a)tdx=(b−a)dt∫ab1(x−a(x−b(x−c)dx=∫01(b−a)(b−a)(b−a)t(b−a)(1−t)(b−a)(1−a+(b−a)t−cb−a)dt=∫011t(1−t)(1−a−cb−a−t)dtx=c+(x−c)t′dx=(x−c)dt′∫c∞1(x−a)(x−b)(x−c)dx=∫0∞(x−c)(x−c)(x−c)t′(x−c)(1−t′)(x−c)dt′=∫0∞1t′(1−t′)(1+x−cc−a−t′)dt′t′=1−tdt′=−dt∫0∞1t′(1−t′)(1+x−cc−a−t′)dt′=∫0−∞−1(1−t)(t)(1+x−cc−a−1+t)dt=∫−∞11(1−t)(t)(1+x−cc−a−1+t)dt=∫−∞11(1−t)(t)(t+x−cc−a)dt=∫−∞11(1−t)(t)(t+b−cb−a)dt=∫011t(1−t)(1−a−cb−a)dt∫abf(x)dx=∫c∞f(x)dx Commented by Ghisom last updated on 02/Nov/24 youwritea+(b−a)tdx=(b−a)dtbutthismakesnosense.pleasecorrectandclarifyyouranswer. Terms of Service Privacy Policy Contact: info@tinkutara.com FacebookTweetPin Post navigation Previous Previous post: lim-n-1-2-n-2-1-2-3-n-2-2-n-n-1-n-2-n-Next Next post: Question-212630 Leave a Reply Cancel replyYour email address will not be published.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.