What is the integral of (sqrt(epi)cos(t)+5/(e+et^2)) ?

The integral of (sqrt(epi)cos(t)+5/(e+et^2)) is sqrt(epi)sin(t)+5/e arctan(t)+C

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

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

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

Solution steps

\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

Apply the Sum Rule:

\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)

Add a constant to the solution

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

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What is the integral of (sqrt(epi)cos(t)+5/(e+et^2)) ?
The integral of (sqrt(epi)cos(t)+5/(e+et^2)) is sqrt(epi)sin(t)+5/e arctan(t)+C