integral of cos(x)sqrt(1+cos^2(x))

Solution

\int cos (x) 1 + cos^{2} (x) d x

arcsin (\frac{1}{2} sin (x)) + \frac{1}{2} sin (2 arcsin (\frac{1}{2} sin (x))) + C

Solution steps

\int cos (x) 1 + cos^{2} (x) d x

Apply Trigonometric Substitution

: \int - u^{2} + 2 d u

= \int - u^{2} + 2 d u

Apply Trigonometric Substitution

: \int 2 cos^{2} (v) d v

= \int 2 cos^{2} (v) d v

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int cos^{2} (v) d v

Rewrite using trig identities

= 2 \cdot \int \frac{1 + cos ( 2 v )}{2} d v

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \frac{1}{2} \cdot \int 1 + cos (2 v) d v

Apply the Sum Rule:

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

= 2 \cdot \frac{1}{2} (\int 1 d v + \int cos (2 v) d v)

\int 1 d v = v

\int cos (2 v) d v = \frac{1}{2} sin (2 v)

= 2 \cdot \frac{1}{2} (v + \frac{1}{2} sin (2 v))

Substitute back

= 2 \cdot \frac{1}{2} (arcsin (\frac{1}{2} sin (x)) + \frac{1}{2} sin (2 arcsin (\frac{1}{2} sin (x))))

Simplify

2 \cdot \frac{1}{2} (arcsin (\frac{1}{2} sin (x)) + \frac{1}{2} sin (2 arcsin (\frac{1}{2} sin (x)))) : arcsin (\frac{1}{2} sin (x)) + \frac{1}{2} sin (2 arcsin (\frac{1}{2} sin (x)))

= arcsin (\frac{1}{2} sin (x)) + \frac{1}{2} sin (2 arcsin (\frac{1}{2} sin (x)))

Add a constant to the solution

= arcsin (\frac{1}{2} sin (x)) + \frac{1}{2} sin (2 arcsin (\frac{1}{2} sin (x))) + C

n = 2 \sum \infty \frac{2}{n ( n - 1 )}

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