let-f-x-1-3-arctan-x-x-t-dt-withx-gt-0-1-determine-a-explicit-form-of-f-x-2-give-f-x-at-form-of-integral-and-find-its-value-3-calculate-1-3-arctan-1-1-t-dt-and-1-3-ar

Question Number 58187 by maxmathsup by imad last updated on 19/Apr/19

l e t f (x) = \int_{1}^{3} a r c t a n (x + \frac{x}{t}) d t w i t h x > 0

1) d e t e r m i n e a e x p l i c i t f o r m o f f (x)

2) g i v e f^{^{'}} (x) a t f o r m o f i n t e g r a l a n d f i n d i t s v a l u e

3) c a l c u l a t e \int_{1}^{3} a r c t a n (1 + \frac{1}{t}) d t a n d \int_{1}^{3} a r c t a n (2 + \frac{2}{t}) d t .

4) c a l c u l a t e \int_{1}^{3} (2 t - 1) a r c t a n (1 + \frac{1}{t}) d t .