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How to use it?
- Integral of d{x}:
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Integral of 2/x^4
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Integral of 1/5x
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Integral of x*dx/(x^2+1)
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Integral of x^3*exp(x^2)
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Identical expressions
- three *dx/cos(seven *x)
- 3 multiply by dx divide by co sinus of e of (7 multiply by x)
- three multiply by dx divide by co sinus of e of (seven multiply by x)
- 3dx/cos(7x)
- 3dx/cos7x
- 3*dx divide by cos(7*x)
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What you mean?
- 3*dx/cos(7*x)
- 3*dx/((cos(7)*x))
- 3*dx*7*x/cos(x)
- 3*dx/cos(7*x)
- 3*dx/((cos(7)*x))
- 3*dx*7*x/cos(x)
- 3*dx*7*x/cos(x)
You entered:
3*dx/cos(7*x)
What you mean?
Integral of 3*dx/cos(7*x) dx
The solution
You have entered
[src]
1 / | | 1 | 3*1*-------- dx | cos(7*x) | / 0
$$\int\limits_{0}^{1} 3 \cdot 1 \cdot \frac{1}{\cos{\left(7 x \right)}}\, dx$$
Integral(3*1/cos(7*x), (x, 0, 1))
Detail solution
We have the integral:
/ | | 1 | 1*3*1*-------- dx | cos(7*x) | /
The integrand
1
3*1*--------
cos(7*x)Multiply numerator and denominator by
cos(7*x)
we get
1 3*cos(7*x)
3*1*-------- = ----------
cos(7*x) 2
cos (7*x) Because
sin(a)^2 + cos(a)^2 = 1
then
2 2 cos (7*x) = 1 - sin (7*x)
transform the denominator
3*cos(7*x) 3*cos(7*x) ---------- = ------------- 2 2 cos (7*x) 1 - sin (7*x)
do replacement
u = sin(7*x)
then the integral
/ | | 3*cos(7*x) | ------------- dx | 2 = | 1 - sin (7*x) | /
/ | | 3*cos(7*x) | ------------- dx | 2 = | 1 - sin (7*x) | /
Because du = 7*dx*cos(7*x)
/ | | 3 | ---------- du | / 2\ | 7*\1 - u / | /
Rewrite the integrand
3 3*1/7 / 1 1 \ ---------- = -----*|----- + -----| / 2\ 2 \1 - u 1 + u/ 7*\1 - u /
then
/ /
| |
| 1 | 1
3* | ----- du 3* | ----- du
/ | 1 + u | 1 - u
| | |
| 3 / / =
| ---------- du = ------------- + -------------
| / 2\ 14 14
| 7*\1 - u /
|
/
= -3*log(-1 + u)/14 + 3*log(1 + u)/14
do backward replacement
u = sin(7*x)
The answer
/ | | 1 3*log(-1 + sin(7*x)) 3*log(1 + sin(7*x)) | 1*3*1*-------- dx = - -------------------- + ------------------- + C0 | cos(7*x) 14 14 | /
where C0 is constant, independent of x
The answer (Indefinite)
[src]
/ | | 1 3*log(-1 + sin(7*x)) 3*log(1 + sin(7*x)) | 3*1*-------- dx = C - -------------------- + ------------------- | cos(7*x) 14 14 | /
$$\int 3 \cdot 1 \cdot \frac{1}{\cos{\left(7 x \right)}}\, dx = C - \frac{3 \log{\left(\sin{\left(7 x \right)} - 1 \right)}}{14} + \frac{3 \log{\left(\sin{\left(7 x \right)} + 1 \right)}}{14}$$
The graph
The graph
Use the examples entering the upper and lower limits of integration.