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Identical expressions
- sin^3x/(two +cosx)
- sinus of cubed x divide by (2 plus co sinus of e of x)
- sinus of cubed x divide by (two plus co sinus of e of x)
- sin3x/(2+cosx)
- sin3x/2+cosx
- sin³x/(2+cosx)
- sin to the power of 3x/(2+cosx)
- sin^3x/2+cosx
- sin^3x divide by (2+cosx)
- sin^3x/(2+cosx)dx
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Similar expressions
- sin^3x/(2-cosx)
Integral of sin^3x/(2+cosx) dx
The solution
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[src]
1 / | | 3 | sin (x) | ---------- dx | 2 + cos(x) | / 0
$$\int\limits_{0}^{1} \frac{\sin^{3}{\left(x \right)}}{\cos{\left(x \right)} + 2}\, dx$$
Integral(sin(x)^3/(2 + cos(x)), (x, 0, 1))
The answer (Indefinite)
[src]
/ | 2/x\ / 2/x\\ / 2/x\\ 2/x\ / 2/x\\ 4/x\ / 2/x\\ 4/x\ / 2/x\\ 2/x\ / 2/x\\ | 3 6*tan |-| 3*log|1 + tan |-|| 3*log|3 + tan |-|| 6*tan |-|*log|1 + tan |-|| 3*tan |-|*log|1 + tan |-|| 3*tan |-|*log|3 + tan |-|| 6*tan |-|*log|3 + tan |-|| | sin (x) 4 \2/ \ \2// \ \2// \2/ \ \2// \2/ \ \2// \2/ \ \2// \2/ \ \2// | ---------- dx = C - ----------------------- - ----------------------- - ----------------------- + ----------------------- - -------------------------- - -------------------------- + -------------------------- + -------------------------- | 2 + cos(x) 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ | 1 + tan |-| + 2*tan |-| 1 + tan |-| + 2*tan |-| 1 + tan |-| + 2*tan |-| 1 + tan |-| + 2*tan |-| 1 + tan |-| + 2*tan |-| 1 + tan |-| + 2*tan |-| 1 + tan |-| + 2*tan |-| 1 + tan |-| + 2*tan |-| / \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/
$$\int \frac{\sin^{3}{\left(x \right)}}{\cos{\left(x \right)} + 2}\, dx = C - \frac{3 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{x}{2} \right)}}{\tan^{4}{\left(\frac{x}{2} \right)} + 2 \tan^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{6 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{4}{\left(\frac{x}{2} \right)} + 2 \tan^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{3 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{4}{\left(\frac{x}{2} \right)} + 2 \tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{3 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 3 \right)} \tan^{4}{\left(\frac{x}{2} \right)}}{\tan^{4}{\left(\frac{x}{2} \right)} + 2 \tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{6 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 3 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{4}{\left(\frac{x}{2} \right)} + 2 \tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{3 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 3 \right)}}{\tan^{4}{\left(\frac{x}{2} \right)} + 2 \tan^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{6 \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{4}{\left(\frac{x}{2} \right)} + 2 \tan^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{4}{\tan^{4}{\left(\frac{x}{2} \right)} + 2 \tan^{2}{\left(\frac{x}{2} \right)} + 1}$$
The graph
The answer
[src]
2 / 2 \ / 2 \ 2 / 2 \ 4 / 2 \ 4 / 2 \ 2 / 2 \
4 6*tan (1/2) 3*log\1 + tan (1/2)/ 3*log\3 + tan (1/2)/ 6*tan (1/2)*log\1 + tan (1/2)/ 3*tan (1/2)*log\1 + tan (1/2)/ 3*tan (1/2)*log\3 + tan (1/2)/ 6*tan (1/2)*log\3 + tan (1/2)/
4 - --------------------------- - 3*log(3) - --------------------------- - --------------------------- + --------------------------- - ------------------------------ - ------------------------------ + ------------------------------ + ------------------------------
4 2 4 2 4 2 4 2 4 2 4 2 4 2 4 2
1 + tan (1/2) + 2*tan (1/2) 1 + tan (1/2) + 2*tan (1/2) 1 + tan (1/2) + 2*tan (1/2) 1 + tan (1/2) + 2*tan (1/2) 1 + tan (1/2) + 2*tan (1/2) 1 + tan (1/2) + 2*tan (1/2) 1 + tan (1/2) + 2*tan (1/2) 1 + tan (1/2) + 2*tan (1/2)
$$- 3 \log{\left(3 \right)} - \frac{4}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{6 \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{3 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{6 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{3 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{1}{2} \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{3 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 3 \right)} \tan^{4}{\left(\frac{1}{2} \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{6 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 3 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{3 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 3 \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} + 4$$
=
=
2 / 2 \ / 2 \ 2 / 2 \ 4 / 2 \ 4 / 2 \ 2 / 2 \
4 6*tan (1/2) 3*log\1 + tan (1/2)/ 3*log\3 + tan (1/2)/ 6*tan (1/2)*log\1 + tan (1/2)/ 3*tan (1/2)*log\1 + tan (1/2)/ 3*tan (1/2)*log\3 + tan (1/2)/ 6*tan (1/2)*log\3 + tan (1/2)/
4 - --------------------------- - 3*log(3) - --------------------------- - --------------------------- + --------------------------- - ------------------------------ - ------------------------------ + ------------------------------ + ------------------------------
4 2 4 2 4 2 4 2 4 2 4 2 4 2 4 2
1 + tan (1/2) + 2*tan (1/2) 1 + tan (1/2) + 2*tan (1/2) 1 + tan (1/2) + 2*tan (1/2) 1 + tan (1/2) + 2*tan (1/2) 1 + tan (1/2) + 2*tan (1/2) 1 + tan (1/2) + 2*tan (1/2) 1 + tan (1/2) + 2*tan (1/2) 1 + tan (1/2) + 2*tan (1/2)
$$- 3 \log{\left(3 \right)} - \frac{4}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{6 \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{3 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{6 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{3 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{1}{2} \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{3 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 3 \right)} \tan^{4}{\left(\frac{1}{2} \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{6 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 3 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{3 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 3 \right)}}{\tan^{4}{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)} + 1} + 4$$
4 - 4/(1 + tan(1/2)^4 + 2*tan(1/2)^2) - 3*log(3) - 6*tan(1/2)^2/(1 + tan(1/2)^4 + 2*tan(1/2)^2) - 3*log(1 + tan(1/2)^2)/(1 + tan(1/2)^4 + 2*tan(1/2)^2) + 3*log(3 + tan(1/2)^2)/(1 + tan(1/2)^4 + 2*tan(1/2)^2) - 6*tan(1/2)^2*log(1 + tan(1/2)^2)/(1 + tan(1/2)^4 + 2*tan(1/2)^2) - 3*tan(1/2)^4*log(1 + tan(1/2)^2)/(1 + tan(1/2)^4 + 2*tan(1/2)^2) + 3*tan(1/2)^4*log(3 + tan(1/2)^2)/(1 + tan(1/2)^4 + 2*tan(1/2)^2) + 6*tan(1/2)^2*log(3 + tan(1/2)^2)/(1 + tan(1/2)^4 + 2*tan(1/2)^2)
The graph
Use the examples entering the upper and lower limits of integration.