👉 Prova ora NerdPal! La nostra nuova app di matematica su iOS e Android
Argomenti Calcolatori
Fogli di lavoro e test
Fogli di lavoro e test color_by_number
diagram_generator Vai a Premium Argomenti
  1. calcolatori
  2. Integrali Di Funzioni Razionali Di Seno E Coseno

Calcolatrice di Integrali di funzioni razionali di seno e coseno

Risolvete i vostri problemi di matematica con la nostra calcolatrice Integrali di funzioni razionali di seno e coseno passo-passo. Migliorate le vostre abilità matematiche con il nostro ampio elenco di problemi impegnativi. Trova tutte le nostre calcolatrici qui.

Modalità simbolica
Modalità testo
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
D□x
∫
∫◻
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

Esempio

Problemi risolti

Problemi difficili

1

Here, we show you a step-by-step solved example of integrals of rational functions of sine and cosine. This solution was automatically generated by our smart calculator:

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

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

Multiplying fractions $\frac{1}{3-\frac{1-t^{2}}{1+t^{2}}} \times \frac{2}{1+t^{2}}$

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

Multiplying the fraction by $-1$

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

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

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

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

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

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

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

Simplifying

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

The integral of a function times a constant ($2$) is equal to the constant times the integral of the function

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

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

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

Subtract the values $3$ and $-1$

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

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

$2\int\frac{1}{4t^{2}+2}dt$
8

Simplify the expression

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

$\sqrt{2t^{2}}$

The power of a product is equal to the product of it's factors raised to the same power

$\sqrt{2}t$
9

Solve the integral applying the substitution $u^2=2t^{2}$. Then, take the square root of both sides, simplifying we have

$u=\sqrt{2}t$

Differentiate both sides of the equation $u=\sqrt{2}t$

$du=\frac{d}{dt}\left(\sqrt{2}t\right)$

Find the derivative

$\frac{d}{dt}\left(\sqrt{2}t\right)$

The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function

$\sqrt{2}\frac{d}{dt}\left(t\right)$

The derivative of the linear function is equal to $1$

$\sqrt{2}$
10

Now, in order to rewrite $dt$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by finding the derivative of the equation above

$du=\sqrt{2}dt$
11

Isolate $dt$ in the previous equation

$\frac{du}{\sqrt{2}}=dt$

Any expression multiplied by $1$ is equal to itself

$2\cdot \left(\frac{\frac{1}{\sqrt{2}}}{2}\right)\int\frac{1}{u^2+1}du$

Simplify the fraction $2\cdot \left(\frac{\frac{1}{\sqrt{2}}}{2}\right)\int\frac{1}{u^2+1}du$

$\frac{1}{\sqrt{2}}\int\frac{1}{u^2+1}du$
12

After replacing everything and simplifying, the integral results in

$\frac{1}{\sqrt{2}}\int\frac{1}{u^2+1}du$
13

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

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

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

$\frac{1}{\sqrt{2}}\arctan\left(\sqrt{2}t\right)$
14

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

$\frac{1}{\sqrt{2}}\arctan\left(\sqrt{2}t\right)$
15

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

$\frac{1}{\sqrt{2}}\arctan\left(\sqrt{2}\tan\left(\frac{x}{2}\right)\right)$
16

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

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

Risposta finale al problema

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

Avete difficoltà in matematica?

Accedete a soluzioni dettagliate passo dopo passo per migliaia di problemi, che crescono ogni giorno!

Calcolatori correlati

Calcolatrice di Integrali trigonometrici

I problemi più diffusi

I problemi più comuni risolti con questa calcolatrice:

$\int\frac{1}{3\cos\left(2x\right)+1}dx$ 85 viste
$\int\left(\frac{1}{2+cos\left(x\right)}\right)dx$ 33 viste
$\int\frac{1}{\cos\left(x\right)+3}dx$ 19 viste
$\int\frac{1}{cos-1}dx$ 10 viste
$\frac{dy}{dx}=1+tan\left(y-x\right)$ 10 viste
$\frac{dy}{dx}=tan\left(x+y-4\right)$ 9 viste
$\int_0^{2\pi}\left(\frac{1}{5+3\cos\left(x\right)}\right)dx$ 8 viste
$\int\left(\frac{1}{1+\tan\left(x\right)}\right)dx$ 8 viste