What is the integral of 1/(5-3cos(x)) ?

The integral of 1/(5-3cos(x)) is 1/2 arctan(2tan(x/2))+C

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integral of 1/(5-3cos(x))

\int \frac{1}{5 - 3 cos ( x )} d x

\frac{1}{2} arctan (2 tan (\frac{x}{2})) + C

Solution steps

\int \frac{1}{5 - 3 cos ( x )} d x

Apply u-substitution

: \int \frac{1}{4 u ^{2} + 1} d u

= \int \frac{1}{4 u ^{2} + 1} d u

Apply Integral Substitution

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

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{1}{v ^{2} + 1} d v

Use the common integral:

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

= \frac{1}{2} arctan (v)

Substitute back

= \frac{1}{2} arctan (2 tan (\frac{x}{2}))

Add a constant to the solution

= \frac{1}{2} arctan (2 tan (\frac{x}{2})) + C

\int 8 e^{4 x} (e^{4 x} - 4)^{\frac{1}{5}} d x

y^{^{'}} + \frac{1}{x} y = 5 x^{2}

2 y^{^{''}} + 26 y = 0, y (0) = 9, y^{^{'}} (0) = 5

\frac{\partial}{\partial x} (9 x^{2} + 25 y^{2} - 225)

\frac{d y}{d x}, 6 x^{3} + 7 y^{3} = 13 x y

What is the integral of 1/(5-3cos(x)) ?
The integral of 1/(5-3cos(x)) is 1/2 arctan(2tan(x/2))+C