Sr Examen
Integral de (sin^6)*5x dx
Solución
Ha introducido
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1 / | | 6 | sin (x)*5*x dx | / 0
$$\int\limits_{0}^{1} x 5 \sin^{6}{\left(x \right)}\, dx$$
Integral((sin(x)^6*5)*x, (x, 0, 1))
Respuesta (Indefinida)
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/ | 6 6 2 4 2 6 2 6 5 3 3 5 2 2 4 2 4 2 | 6 25*cos (x) 265*sin (x) 25*cos (x)*sin (x) 25*x *cos (x) 25*x *sin (x) 55*x*sin (x)*cos(x) 25*x*cos (x)*sin (x) 25*x*cos (x)*sin(x) 75*x *cos (x)*sin (x) 75*x *cos (x)*sin (x) | sin (x)*5*x dx = C - ---------- + ----------- + ------------------ + ------------- + ------------- - ------------------- - -------------------- - ------------------- + --------------------- + --------------------- | 96 288 24 32 32 16 6 16 32 32 /
$$\int x 5 \sin^{6}{\left(x \right)}\, dx = C + \frac{25 x^{2} \sin^{6}{\left(x \right)}}{32} + \frac{75 x^{2} \sin^{4}{\left(x \right)} \cos^{2}{\left(x \right)}}{32} + \frac{75 x^{2} \sin^{2}{\left(x \right)} \cos^{4}{\left(x \right)}}{32} + \frac{25 x^{2} \cos^{6}{\left(x \right)}}{32} - \frac{55 x \sin^{5}{\left(x \right)} \cos{\left(x \right)}}{16} - \frac{25 x \sin^{3}{\left(x \right)} \cos^{3}{\left(x \right)}}{6} - \frac{25 x \sin{\left(x \right)} \cos^{5}{\left(x \right)}}{16} + \frac{265 \sin^{6}{\left(x \right)}}{288} + \frac{25 \sin^{4}{\left(x \right)} \cos^{2}{\left(x \right)}}{24} - \frac{25 \cos^{6}{\left(x \right)}}{96}$$
Gráfica
Respuesta
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6 6 5 3 3 5 2 4 4 2 175 25*cos (1) 65*sin (1) 55*sin (1)*cos(1) 25*cos (1)*sin (1) 25*cos (1)*sin(1) 75*cos (1)*sin (1) 125*cos (1)*sin (1) --- + ---------- + ---------- - ----------------- - ------------------ - ----------------- + ------------------ + ------------------- 288 144 48 16 6 16 32 96
$$- \frac{55 \sin^{5}{\left(1 \right)} \cos{\left(1 \right)}}{16} - \frac{25 \sin^{3}{\left(1 \right)} \cos^{3}{\left(1 \right)}}{6} - \frac{25 \sin{\left(1 \right)} \cos^{5}{\left(1 \right)}}{16} + \frac{25 \cos^{6}{\left(1 \right)}}{144} + \frac{125 \sin^{2}{\left(1 \right)} \cos^{4}{\left(1 \right)}}{96} + \frac{75 \sin^{4}{\left(1 \right)} \cos^{2}{\left(1 \right)}}{32} + \frac{65 \sin^{6}{\left(1 \right)}}{48} + \frac{175}{288}$$
=
=
6 6 5 3 3 5 2 4 4 2 175 25*cos (1) 65*sin (1) 55*sin (1)*cos(1) 25*cos (1)*sin (1) 25*cos (1)*sin(1) 75*cos (1)*sin (1) 125*cos (1)*sin (1) --- + ---------- + ---------- - ----------------- - ------------------ - ----------------- + ------------------ + ------------------- 288 144 48 16 6 16 32 96
$$- \frac{55 \sin^{5}{\left(1 \right)} \cos{\left(1 \right)}}{16} - \frac{25 \sin^{3}{\left(1 \right)} \cos^{3}{\left(1 \right)}}{6} - \frac{25 \sin{\left(1 \right)} \cos^{5}{\left(1 \right)}}{16} + \frac{25 \cos^{6}{\left(1 \right)}}{144} + \frac{125 \sin^{2}{\left(1 \right)} \cos^{4}{\left(1 \right)}}{96} + \frac{75 \sin^{4}{\left(1 \right)} \cos^{2}{\left(1 \right)}}{32} + \frac{65 \sin^{6}{\left(1 \right)}}{48} + \frac{175}{288}$$
175/288 + 25*cos(1)^6/144 + 65*sin(1)^6/48 - 55*sin(1)^5*cos(1)/16 - 25*cos(1)^3*sin(1)^3/6 - 25*cos(1)^5*sin(1)/16 + 75*cos(1)^2*sin(1)^4/32 + 125*cos(1)^4*sin(1)^2/96
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.