integral of (sqrt(epi)cos(t)+5/(e+et^2))

解

\int (e π cos (t) + \frac{5}{e + e t ^{2}}) d t

e π sin (t) + \frac{5}{e} arctan (t) + C

解答ステップ

\int e π cos (t) + \frac{5}{e + e t ^{2}} d t

\frac{5}{e + e t ^{2}} = \frac{5}{e} \frac{1}{1 + t ^{2}}

= \int e π cos (t) + \frac{5}{e} \cdot \frac{1}{1 + t ^{2}} d t

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int e π cos (t) d t + \int \frac{5}{e} \cdot \frac{1}{1 + t ^{2}} d t

\int e π cos (t) d t = e π sin (t)

\int \frac{5}{e} \cdot \frac{1}{1 + t ^{2}} d t = \frac{5}{e} arctan (t)

= e π sin (t) + \frac{5}{e} arctan (t)

定数を解答に追加する

= e π sin (t) + \frac{5}{e} arctan (t) + C