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How to use it?
- Integral of d{x}:
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Integral of x^2/x
- Integral of e^x/(1+x)
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Integral of x^2/(√(1-x^2))
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Integral of sin3
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Identical expressions
- one /sin(three *x)
- 1 divide by sinus of (3 multiply by x)
- one divide by sinus of (three multiply by x)
- 1/sin(3x)
- 1/sin3x
- 1 divide by sin(3*x)
- 1/sin(3*x)dx
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Similar expressions
- 1/(sin(3*x)+4)
- 1/(sin^3(x)+cos^3(x))
- 1/(sin3x+6)
- 1/(((sin3x)^2)*((1/tg3x)))
- 1/(sin(3x)^2*cos(3x))
Integral of 1/sin(3*x) dx
The solution
You have entered
[src]
1 / | | 1 | -------- dx | sin(3*x) | / 0
$$\int\limits_{0}^{1} \frac{1}{\sin{\left(3 x \right)}}\, dx$$
Integral(1/sin(3*x), (x, 0, 1))
Detail solution
We have the integral:
/ | | 1 | -------- dx | sin(3*x) | /
The integrand
1 -------- sin(3*x)
Multiply numerator and denominator by
sin(3*x)
we get
1 sin(3*x)
-------- = ---------
sin(3*x) 2
sin (3*x)Because
sin(a)^2 + cos(a)^2 = 1
then
2 2 sin (3*x) = 1 - cos (3*x)
transform the denominator
sin(3*x) sin(3*x) --------- = ------------- 2 2 sin (3*x) 1 - cos (3*x)
do replacement
u = cos(3*x)
then the integral
/ | | sin(3*x) | ------------- dx | 2 = | 1 - cos (3*x) | /
/ | | sin(3*x) | ------------- dx | 2 = | 1 - cos (3*x) | /
Because du = -3*dx*sin(3*x)
/ | | -1 | ---------- du | / 2\ | 3*\1 - u / | /
Rewrite the integrand
-1 -1 / 1 1 \ ---------- = ---*|----- + -----| / 2\ 3*2 \1 - u 1 + u/ 3*\1 - u /
then
/ /
| |
| 1 | 1
| ----- du | ----- du
/ | 1 + u | 1 - u
| | |
| -1 / / =
| ---------- du = - ----------- - -----------
| / 2\ 6 6
| 3*\1 - u /
|
/
= -log(1 + u)/6 + log(-1 + u)/6
do backward replacement
u = cos(3*x)
The answer
/ | | 1 log(1 + cos(3*x)) log(-1 + cos(3*x)) | -------- dx = - ----------------- + ------------------ + C0 | sin(3*x) 6 6 | /
where C0 is constant, independent of x
The answer (Indefinite)
[src]
/ | | 1 log(1 + cos(3*x)) log(-1 + cos(3*x)) | -------- dx = C - ----------------- + ------------------ | sin(3*x) 6 6 | /
$$\int \frac{1}{\sin{\left(3 x \right)}}\, dx = C + \frac{\log{\left(\cos{\left(3 x \right)} - 1 \right)}}{6} - \frac{\log{\left(\cos{\left(3 x \right)} + 1 \right)}}{6}$$
The graph
The answer
[src]
pi*I
oo + ----
6
$$\infty + \frac{i \pi}{6}$$
=
=
pi*I
oo + ----
6
$$\infty + \frac{i \pi}{6}$$
oo + pi*i/6
Use the examples entering the upper and lower limits of integration.