Sr Examen
Integral de (sin^6)(x/3) dx
Solución
Ha introducido
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1 / | | 6 x | sin (x)*- dx | 3 | / 0
$$\int\limits_{0}^{1} \frac{x}{3} \sin^{6}{\left(x \right)}\, dx$$
Integral(sin(x)^6*(x/3), (x, 0, 1))
Respuesta (Indefinida)
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/ | 6 6 4 2 2 6 2 6 5 3 3 5 2 2 4 2 4 2 | 6 x 35*cos (x) 11*sin (x) 5*cos (x)*sin (x) 5*x *cos (x) 5*x *sin (x) 11*x*sin (x)*cos(x) 5*x*cos (x)*sin (x) 5*x*cos (x)*sin(x) 5*x *cos (x)*sin (x) 5*x *cos (x)*sin (x) | sin (x)*- dx = C - ---------- + ---------- - ----------------- + ------------ + ------------ - ------------------- - ------------------- - ------------------ + -------------------- + -------------------- | 3 864 288 72 96 96 48 18 48 32 32 | /
$$\int \frac{x}{3} \sin^{6}{\left(x \right)}\, dx = C + \frac{5 x^{2} \sin^{6}{\left(x \right)}}{96} + \frac{5 x^{2} \sin^{4}{\left(x \right)} \cos^{2}{\left(x \right)}}{32} + \frac{5 x^{2} \sin^{2}{\left(x \right)} \cos^{4}{\left(x \right)}}{32} + \frac{5 x^{2} \cos^{6}{\left(x \right)}}{96} - \frac{11 x \sin^{5}{\left(x \right)} \cos{\left(x \right)}}{48} - \frac{5 x \sin^{3}{\left(x \right)} \cos^{3}{\left(x \right)}}{18} - \frac{5 x \sin{\left(x \right)} \cos^{5}{\left(x \right)}}{48} + \frac{11 \sin^{6}{\left(x \right)}}{288} - \frac{5 \sin^{2}{\left(x \right)} \cos^{4}{\left(x \right)}}{72} - \frac{35 \cos^{6}{\left(x \right)}}{864}$$
Gráfica
Respuesta
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6 6 5 3 3 5 2 4 4 2 35 5*cos (1) 13*sin (1) 11*sin (1)*cos(1) 5*cos (1)*sin (1) 5*cos (1)*sin(1) 5*cos (1)*sin (1) 25*cos (1)*sin (1) --- + --------- + ---------- - ----------------- - ----------------- - ---------------- + ----------------- + ------------------ 864 432 144 48 18 48 32 288
$$- \frac{11 \sin^{5}{\left(1 \right)} \cos{\left(1 \right)}}{48} - \frac{5 \sin^{3}{\left(1 \right)} \cos^{3}{\left(1 \right)}}{18} - \frac{5 \sin{\left(1 \right)} \cos^{5}{\left(1 \right)}}{48} + \frac{5 \cos^{6}{\left(1 \right)}}{432} + \frac{25 \sin^{2}{\left(1 \right)} \cos^{4}{\left(1 \right)}}{288} + \frac{5 \sin^{4}{\left(1 \right)} \cos^{2}{\left(1 \right)}}{32} + \frac{13 \sin^{6}{\left(1 \right)}}{144} + \frac{35}{864}$$
=
=
6 6 5 3 3 5 2 4 4 2 35 5*cos (1) 13*sin (1) 11*sin (1)*cos(1) 5*cos (1)*sin (1) 5*cos (1)*sin(1) 5*cos (1)*sin (1) 25*cos (1)*sin (1) --- + --------- + ---------- - ----------------- - ----------------- - ---------------- + ----------------- + ------------------ 864 432 144 48 18 48 32 288
$$- \frac{11 \sin^{5}{\left(1 \right)} \cos{\left(1 \right)}}{48} - \frac{5 \sin^{3}{\left(1 \right)} \cos^{3}{\left(1 \right)}}{18} - \frac{5 \sin{\left(1 \right)} \cos^{5}{\left(1 \right)}}{48} + \frac{5 \cos^{6}{\left(1 \right)}}{432} + \frac{25 \sin^{2}{\left(1 \right)} \cos^{4}{\left(1 \right)}}{288} + \frac{5 \sin^{4}{\left(1 \right)} \cos^{2}{\left(1 \right)}}{32} + \frac{13 \sin^{6}{\left(1 \right)}}{144} + \frac{35}{864}$$
35/864 + 5*cos(1)^6/432 + 13*sin(1)^6/144 - 11*sin(1)^5*cos(1)/48 - 5*cos(1)^3*sin(1)^3/18 - 5*cos(1)^5*sin(1)/48 + 5*cos(1)^2*sin(1)^4/32 + 25*cos(1)^4*sin(1)^2/288
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.