What is the integral of 1/(5-4sin(x)+3cos(x)) ?

The integral of 1/(5-4sin(x)+3cos(x)) is -1/(tan(x/2)-2)+C

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integral of 1/(5-4sin(x)+3cos(x))

\int \frac{1}{5 - 4 sin ( x ) + 3 cos ( x )} d x

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

Solution steps

\int \frac{1}{5 - 4 sin ( x ) + 3 cos ( x )} d x

Apply u-substitution

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

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

Complete the square

u^{2} - 4 u + 4 : (u - 2)^{2}

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

Apply u-substitution

: \int \frac{1}{v ^{2}} d v

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

Apply the Power Rule

: - \frac{1}{v}

= - \frac{1}{v}

Substitute back

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

Add a constant to the solution

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

- 4 y^{^{'}} + y = 1

e^{x}, - x + 1, x = 1

\frac{d}{d x} ((2 x + 5)^{2})

\int_{- 2}^{1} (x - 8 ∣ x ∣) d x

What is the integral of 1/(5-4sin(x)+3cos(x)) ?
The integral of 1/(5-4sin(x)+3cos(x)) is -1/(tan(x/2)-2)+C