What is the integral of 1/(5+4sec(x)) ?

The integral of 1/(5+4sec(x)) is 2(-2/15 ln|tan(x/2)+3|+2/15 ln|tan(x/2)-3|+1/5 arctan(tan(x/2)))+C

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integral of 1/(5+4sec(x))

\int \frac{1}{5 + 4 sec ( x )} d x

2 (- \frac{2}{15} ln tan (\frac{x}{2}) + 3 + \frac{2}{15} ln tan (\frac{x}{2}) - 3 + \frac{1}{5} arctan (tan (\frac{x}{2}))) + C

Solution steps

\int \frac{1}{5 + 4 sec ( x )} d x

Rewrite using trig identities

= \int \frac{cos ( x )}{5 cos ( x ) + 4} d x

Apply u-substitution

: \int \frac{2 ( 1 - u ^{2} )}{( - u ^{2} + 9 ) ( 1 + u ^{2} )} d u

= \int \frac{2 ( 1 - u ^{2} )}{( - u ^{2} + 9 ) ( 1 + u ^{2} )} d u

Take the constant out:

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

= 2 \cdot \int \frac{1 - u ^{2}}{( - u ^{2} + 9 ) ( 1 + u ^{2} )} d u

Take the partial fraction of

\frac{1 - u ^{2}}{( - u ^{2} + 9 ) ( 1 + u ^{2} )} : - \frac{2}{15 ( u + 3 )} + \frac{2}{15 ( u - 3 )} + \frac{1}{5 ( u ^{2} + 1 )}

= 2 \cdot \int - \frac{2}{15 ( u + 3 )} + \frac{2}{15 ( u - 3 )} + \frac{1}{5 ( u ^{2} + 1 )} d u

Apply the Sum Rule:

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

= 2 (- \int \frac{2}{15 ( u + 3 )} d u + \int \frac{2}{15 ( u - 3 )} d u + \int \frac{1}{5 ( u ^{2} + 1 )} d u)

\int \frac{2}{15 ( u + 3 )} d u = \frac{2}{15} ln ∣ u + 3 ∣

\int \frac{2}{15 ( u - 3 )} d u = \frac{2}{15} ln ∣ u - 3 ∣

\int \frac{1}{5 ( u ^{2} + 1 )} d u = \frac{1}{5} arctan (u)

= 2 (- \frac{2}{15} ln ∣ u + 3 ∣ + \frac{2}{15} ln ∣ u - 3 ∣ + \frac{1}{5} arctan (u))

Substitute back

u = tan (\frac{x}{2})

= 2 (- \frac{2}{15} ln tan (\frac{x}{2}) + 3 + \frac{2}{15} ln tan (\frac{x}{2}) - 3 + \frac{1}{5} arctan (tan (\frac{x}{2})))

Add a constant to the solution

= 2 (- \frac{2}{15} ln tan (\frac{x}{2}) + 3 + \frac{2}{15} ln tan (\frac{x}{2}) - 3 + \frac{1}{5} arctan (tan (\frac{x}{2}))) + C

y^{^{'}} + \frac{y}{x} = 2 y^{2}

\frac{d}{d x} (\frac{4 x + 2}{( 3 - 2 x ) ^{3}})

\frac{\partial}{\partial x} (x e^{14 x y})

y^{^{'}} = 3 - \frac{3 x}{10000}

\int t^{2} e^{2 t} d t

What is the integral of 1/(5+4sec(x)) ?
The integral of 1/(5+4sec(x)) is 2(-2/15 ln|tan(x/2)+3|+2/15 ln|tan(x/2)-3|+1/5 arctan(tan(x/2)))+C