# Integrals of Rational Functions of Sine and Cosine Calculator

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### Difficult Problems

1

Solved example of integrals of rational functions of sine and cosine

$\int\frac{dx}{3-cos\left(x\right)}$
2

We can solve the integral $\int\frac{1}{3-\cos\left(x\right)}dx$ by applying the method Weierstrass substitution (also known as tangent half-angle substitution) which converts an integral of trigonometric functions into a rational function of $t$ by setting the substitution

$t=\tan\left(\frac{x}{2}\right)$
3

Hence

$\sin x=\frac{2t}{1+t^{2}},\:\cos x=\frac{1-t^{2}}{1+t^{2}},\:\mathrm{and}\:\:dx=\frac{2}{1+t^{2}}dt$
4

Substituting in the original integral we get

$\int\frac{1}{3-\left(\frac{1-t^{2}}{1+t^{2}}\right)}\frac{2}{1+t^{2}}dt$

Combine $3-\left(\frac{1-t^{2}}{1+t^{2}}\right)$ in a single fraction

$\int\frac{1+t^{2}}{-\left(1-t^{2}\right)+3\left(1+t^{2}\right)}\frac{2}{1+t^{2}}dt$

Multiplying fractions $\frac{1+t^{2}}{-\left(1-t^{2}\right)+3\left(1+t^{2}\right)} \times \frac{2}{1+t^{2}}$

$\int\frac{2\left(1+t^{2}\right)}{\left(-\left(1-t^{2}\right)+3\left(1+t^{2}\right)\right)\left(1+t^{2}\right)}dt$

Simplify the fraction $\frac{2\left(1+t^{2}\right)}{\left(-\left(1-t^{2}\right)+3\left(1+t^{2}\right)\right)\left(1+t^{2}\right)}$ by $1+t^{2}$

$\int\frac{2}{-\left(1-t^{2}\right)+3\left(1+t^{2}\right)}dt$
5

Simplifying

$\int\frac{2}{-\left(1-t^{2}\right)+3\left(1+t^{2}\right)}dt$
6

Solve the product $-\left(1-t^{2}\right)$

$\int\frac{2}{-1+t^{2}+3\left(1+t^{2}\right)}dt$
7

Solve the product $3\left(1+t^{2}\right)$

$\int\frac{2}{2+t^{2}+3t^{2}}dt$
8

Combining like terms $t^{2}$ and $3t^{2}$

$\int\frac{2}{2+4t^{2}}dt$
9

The integral of a constant by a function is equal to the constant multiplied by the integral of the function

$2\int\frac{1}{2+4t^{2}}dt$
10

Solve the integral applying the substitution $u^2=2t^{2}$

$\sqrt[4]{\frac{1}{4}}\int\frac{1}{1+u^2}du$
11

Solve the integral applying the formula $\displaystyle\int\frac{x'}{x^2+a^2}dx=\frac{1}{a}\arctan\left(\frac{x}{a}\right)$

$\sqrt[4]{\frac{1}{4}}\arctan\left(u\right)$

$\sqrt[4]{\frac{1}{4}}\arctan\left(\sqrt[4]{4}t\right)$
12

Replace $u$ with the value that we assigned to it in the beginning: $\sqrt[4]{4}t$

$\sqrt[4]{\frac{1}{4}}\arctan\left(\sqrt[4]{4}t\right)$
13

Replace $t$ with the value that we assigned to it in the beginning: $\tan\left(\frac{x}{2}\right)$

$\sqrt[4]{\frac{1}{4}}\arctan\left(\sqrt[4]{4}\tan\left(\frac{x}{2}\right)\right)$
14

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

$\sqrt[4]{\frac{1}{4}}\arctan\left(\sqrt[4]{4}\tan\left(\frac{x}{2}\right)\right)+C_0$

$\sqrt[4]{\frac{1}{4}}\arctan\left(\sqrt[4]{4}\tan\left(\frac{x}{2}\right)\right)+C_0$