1. calculators
  2. Integrals Of Rational Functions Of Sine And Cosine

Integrals of Rational Functions of Sine and Cosine Calculator

Get detailed solutions to your math problems with our Integrals of Rational Functions of Sine and Cosine step-by-step calculator. Practice your math skills and learn step by step with our math solver. Check out all of our online calculators here!

Go!
1
2
3
4
5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

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

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

Divide fractions $\frac{1}{\frac{-\left(1-t^{2}\right)+3\left(1+t^{2}\right)}{1+t^{2}}}$ with Keep, Change, Flip: $a\div \frac{b}{c}=\frac{a}{1}\div\frac{b}{c}=\frac{a}{1}\times\frac{c}{b}=\frac{a\cdot c}{b}$

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

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

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

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

Multiply $-1$ times $-1$

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

Any expression multiplied by $1$ is equal to itself

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

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

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

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

Multiply $-1$ times $-1$

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

Any expression multiplied by $1$ is equal to itself

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

$\int\frac{2}{-1+t^{2}+3\cdot 1+3t^{2}}dt$

Multiply $3$ times $1$

$\int\frac{2}{-1+t^{2}+3+3t^{2}}dt$

Subtract the values $3$ and $-1$

$\int\frac{2}{2+t^{2}+3t^{2}}dt$
7

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

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

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

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

Factor the denominator by $2$

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

Cancel the common factor $2$

$\int\frac{1}{1+2t^{2}}dt$
8

Simplifying

$\int\frac{1}{1+2t^{2}}dt$
9

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

$\frac{\sqrt{2}}{2}\int\frac{1}{1+u^2}du$

$\frac{\sqrt{2}}{2}\left(\frac{1}{\sqrt{1}}\right)\arctan\left(\frac{u}{\sqrt{1}}\right)$

Calculate the power $\sqrt{1}$

$\frac{\sqrt{2}}{2}\left(\frac{1}{1}\right)\arctan\left(\frac{u}{\sqrt{1}}\right)$

Any expression divided by one ($1$) is equal to that same expression

$\frac{\sqrt{2}}{2}\arctan\left(u\right)$
10

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

$\frac{\sqrt{2}}{2}\arctan\left(u\right)$

$\frac{\sqrt{2}}{2}\arctan\left(\frac{2}{\sqrt{2}}t\right)$
11

Replace $u$ with the value that we assigned to it in the beginning: $\frac{2}{\sqrt{2}}t$

$\frac{\sqrt{2}}{2}\arctan\left(\frac{2}{\sqrt{2}}t\right)$
12

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

$\frac{\sqrt{2}}{2}\arctan\left(\frac{2}{\sqrt{2}}\tan\left(\frac{x}{2}\right)\right)$
13

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

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

Final Answer

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

Struggling with math?

Access detailed step by step solutions to thousands of problems, growing every day!