Sr Examen
Integral de cos(x-pi/3)^2 dx
Solución
Ha introducido
[src]
1 / | | 2/ pi\ | cos |x - --| dx | \ 3 / | / 0
$$\int\limits_{0}^{1} \cos^{2}{\left(x - \frac{\pi}{3} \right)}\, dx$$
Integral(cos(x - pi/3)^2, (x, 0, 1))
Respuesta (Indefinida)
[src]
/ /x pi\ 3/x pi\ 4/x pi\ 2/x pi\ | 2*cot|- + --| 2*cot |- + --| x*cot |- + --| 2*x*cot |- + --| | 2/ pi\ x \2 3 / \2 3 / \2 3 / \2 3 / | cos |x - --| dx = C + ----------------------------------- - ----------------------------------- + ----------------------------------- + ----------------------------------- + ----------------------------------- | \ 3 / 4/x pi\ 2/x pi\ 4/x pi\ 2/x pi\ 4/x pi\ 2/x pi\ 4/x pi\ 2/x pi\ 4/x pi\ 2/x pi\ | 2 + 2*cot |- + --| + 4*cot |- + --| 2 + 2*cot |- + --| + 4*cot |- + --| 2 + 2*cot |- + --| + 4*cot |- + --| 2 + 2*cot |- + --| + 4*cot |- + --| 2 + 2*cot |- + --| + 4*cot |- + --| / \2 3 / \2 3 / \2 3 / \2 3 / \2 3 / \2 3 / \2 3 / \2 3 / \2 3 / \2 3 /
$$\int \cos^{2}{\left(x - \frac{\pi}{3} \right)}\, dx = C + \frac{x \cot^{4}{\left(\frac{x}{2} + \frac{\pi}{3} \right)}}{2 \cot^{4}{\left(\frac{x}{2} + \frac{\pi}{3} \right)} + 4 \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{3} \right)} + 2} + \frac{2 x \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{3} \right)}}{2 \cot^{4}{\left(\frac{x}{2} + \frac{\pi}{3} \right)} + 4 \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{3} \right)} + 2} + \frac{x}{2 \cot^{4}{\left(\frac{x}{2} + \frac{\pi}{3} \right)} + 4 \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{3} \right)} + 2} + \frac{2 \cot^{3}{\left(\frac{x}{2} + \frac{\pi}{3} \right)}}{2 \cot^{4}{\left(\frac{x}{2} + \frac{\pi}{3} \right)} + 4 \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{3} \right)} + 2} - \frac{2 \cot{\left(\frac{x}{2} + \frac{\pi}{3} \right)}}{2 \cot^{4}{\left(\frac{x}{2} + \frac{\pi}{3} \right)} + 4 \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{3} \right)} + 2}$$
Gráfica
Respuesta
[src]
2/ pi\ 2/ pi\ / pi\ / pi\
cos |1 + --| sin |1 + --| ___ cos|1 + --|*sin|1 + --|
\ 6 / \ 6 / \/ 3 \ 6 / \ 6 /
------------ + ------------ + ----- - -----------------------
2 2 8 2
$$- \frac{\sin{\left(\frac{\pi}{6} + 1 \right)} \cos{\left(\frac{\pi}{6} + 1 \right)}}{2} + \frac{\cos^{2}{\left(\frac{\pi}{6} + 1 \right)}}{2} + \frac{\sqrt{3}}{8} + \frac{\sin^{2}{\left(\frac{\pi}{6} + 1 \right)}}{2}$$
=
=
2/ pi\ 2/ pi\ / pi\ / pi\
cos |1 + --| sin |1 + --| ___ cos|1 + --|*sin|1 + --|
\ 6 / \ 6 / \/ 3 \ 6 / \ 6 /
------------ + ------------ + ----- - -----------------------
2 2 8 2
$$- \frac{\sin{\left(\frac{\pi}{6} + 1 \right)} \cos{\left(\frac{\pi}{6} + 1 \right)}}{2} + \frac{\cos^{2}{\left(\frac{\pi}{6} + 1 \right)}}{2} + \frac{\sqrt{3}}{8} + \frac{\sin^{2}{\left(\frac{\pi}{6} + 1 \right)}}{2}$$
cos(1 + pi/6)^2/2 + sin(1 + pi/6)^2/2 + sqrt(3)/8 - cos(1 + pi/6)*sin(1 + pi/6)/2
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.