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Identical expressions
- r*cosr^ two
- r multiply by co sinus of e of r squared
- r multiply by co sinus of e of r to the power of two
- r*cosr2
- r*cosr²
- r*cosr to the power of 2
- rcosr^2
- rcosr2
Integral of r*cosr^2 dr
The solution
You have entered
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___ ____
\/ 2 *\/ pi
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2
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| r*cos (r) dr
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\/ 3 *\/ pi
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3
$$\int\limits_{\frac{\sqrt{3} \sqrt{\pi}}{3}}^{\frac{\sqrt{2} \sqrt{\pi}}{2}} r \cos^{2}{\left(r \right)}\, dr$$
Integral(r*cos(r)^2, (r, sqrt(3)*sqrt(pi)/3, sqrt(2)*sqrt(pi)/2))
The answer (Indefinite)
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/ | 2 2 2 2 2 | 2 sin (r) r *cos (r) r *sin (r) r*cos(r)*sin(r) | r*cos (r) dr = C - ------- + ---------- + ---------- + --------------- | 4 4 4 2 /
$$\int r \cos^{2}{\left(r \right)}\, dr = C + \frac{r^{2} \sin^{2}{\left(r \right)}}{4} + \frac{r^{2} \cos^{2}{\left(r \right)}}{4} + \frac{r \sin{\left(r \right)} \cos{\left(r \right)}}{2} - \frac{\sin^{2}{\left(r \right)}}{4}$$
The graph
The answer
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/ ___ ____\ / ___ ____\ / ___ ____\ / ___ ____\ / ___ ____\ / ___ ____\ / ___ ____\ / ___ ____\ / ___ ____\ / ___ ____\
2|\/ 2 *\/ pi | 2|\/ 3 *\/ pi | 2|\/ 3 *\/ pi | 2|\/ 3 *\/ pi | 2|\/ 2 *\/ pi | 2|\/ 2 *\/ pi | ___ ____ |\/ 3 *\/ pi | |\/ 3 *\/ pi | ___ ____ |\/ 2 *\/ pi | |\/ 2 *\/ pi |
sin |------------| sin |------------| pi*cos |------------| pi*sin |------------| pi*cos |------------| pi*sin |------------| \/ 3 *\/ pi *cos|------------|*sin|------------| \/ 2 *\/ pi *cos|------------|*sin|------------|
\ 2 / \ 3 / \ 3 / \ 3 / \ 2 / \ 2 / \ 3 / \ 3 / \ 2 / \ 2 /
- ------------------ + ------------------ - --------------------- - --------------------- + --------------------- + --------------------- - ------------------------------------------------ + ------------------------------------------------
4 4 12 12 8 8 6 4
$$- \frac{\sqrt{3} \sqrt{\pi} \sin{\left(\frac{\sqrt{3} \sqrt{\pi}}{3} \right)} \cos{\left(\frac{\sqrt{3} \sqrt{\pi}}{3} \right)}}{6} - \frac{\sin^{2}{\left(\frac{\sqrt{2} \sqrt{\pi}}{2} \right)}}{4} - \frac{\pi \sin^{2}{\left(\frac{\sqrt{3} \sqrt{\pi}}{3} \right)}}{12} - \frac{\pi \cos^{2}{\left(\frac{\sqrt{3} \sqrt{\pi}}{3} \right)}}{12} + \frac{\pi \cos^{2}{\left(\frac{\sqrt{2} \sqrt{\pi}}{2} \right)}}{8} + \frac{\sin^{2}{\left(\frac{\sqrt{3} \sqrt{\pi}}{3} \right)}}{4} + \frac{\sqrt{2} \sqrt{\pi} \sin{\left(\frac{\sqrt{2} \sqrt{\pi}}{2} \right)} \cos{\left(\frac{\sqrt{2} \sqrt{\pi}}{2} \right)}}{4} + \frac{\pi \sin^{2}{\left(\frac{\sqrt{2} \sqrt{\pi}}{2} \right)}}{8}$$
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/ ___ ____\ / ___ ____\ / ___ ____\ / ___ ____\ / ___ ____\ / ___ ____\ / ___ ____\ / ___ ____\ / ___ ____\ / ___ ____\
2|\/ 2 *\/ pi | 2|\/ 3 *\/ pi | 2|\/ 3 *\/ pi | 2|\/ 3 *\/ pi | 2|\/ 2 *\/ pi | 2|\/ 2 *\/ pi | ___ ____ |\/ 3 *\/ pi | |\/ 3 *\/ pi | ___ ____ |\/ 2 *\/ pi | |\/ 2 *\/ pi |
sin |------------| sin |------------| pi*cos |------------| pi*sin |------------| pi*cos |------------| pi*sin |------------| \/ 3 *\/ pi *cos|------------|*sin|------------| \/ 2 *\/ pi *cos|------------|*sin|------------|
\ 2 / \ 3 / \ 3 / \ 3 / \ 2 / \ 2 / \ 3 / \ 3 / \ 2 / \ 2 /
- ------------------ + ------------------ - --------------------- - --------------------- + --------------------- + --------------------- - ------------------------------------------------ + ------------------------------------------------
4 4 12 12 8 8 6 4
$$- \frac{\sqrt{3} \sqrt{\pi} \sin{\left(\frac{\sqrt{3} \sqrt{\pi}}{3} \right)} \cos{\left(\frac{\sqrt{3} \sqrt{\pi}}{3} \right)}}{6} - \frac{\sin^{2}{\left(\frac{\sqrt{2} \sqrt{\pi}}{2} \right)}}{4} - \frac{\pi \sin^{2}{\left(\frac{\sqrt{3} \sqrt{\pi}}{3} \right)}}{12} - \frac{\pi \cos^{2}{\left(\frac{\sqrt{3} \sqrt{\pi}}{3} \right)}}{12} + \frac{\pi \cos^{2}{\left(\frac{\sqrt{2} \sqrt{\pi}}{2} \right)}}{8} + \frac{\sin^{2}{\left(\frac{\sqrt{3} \sqrt{\pi}}{3} \right)}}{4} + \frac{\sqrt{2} \sqrt{\pi} \sin{\left(\frac{\sqrt{2} \sqrt{\pi}}{2} \right)} \cos{\left(\frac{\sqrt{2} \sqrt{\pi}}{2} \right)}}{4} + \frac{\pi \sin^{2}{\left(\frac{\sqrt{2} \sqrt{\pi}}{2} \right)}}{8}$$
-sin(sqrt(2)*sqrt(pi)/2)^2/4 + sin(sqrt(3)*sqrt(pi)/3)^2/4 - pi*cos(sqrt(3)*sqrt(pi)/3)^2/12 - pi*sin(sqrt(3)*sqrt(pi)/3)^2/12 + pi*cos(sqrt(2)*sqrt(pi)/2)^2/8 + pi*sin(sqrt(2)*sqrt(pi)/2)^2/8 - sqrt(3)*sqrt(pi)*cos(sqrt(3)*sqrt(pi)/3)*sin(sqrt(3)*sqrt(pi)/3)/6 + sqrt(2)*sqrt(pi)*cos(sqrt(2)*sqrt(pi)/2)*sin(sqrt(2)*sqrt(pi)/2)/4
The graph
Use the examples entering the upper and lower limits of integration.