Bài 2 trang 112 SGK Giải tích 12

Giải bài 2 trang 112 SGK Giải tích 12.Tính các tích phân.

Đề bài

Tính các tích phân sau:

a) (int_0^2 {left| {1 - x} ight|} dx)                          b) (int_0^{{pi  over 2}} s i{n^2}xdx)

c) (int_0^{ln2} {{{{e^{2x + 1}} + 1} over {{e^x}}}} dx)                       d) (int_0^pi  s in2xco{s^2}xdx)

Lời giải chi tiết

a) Ta có: (left| {1 - x} ight| = left[ egin{array}{l}1 - x,,khi,,x le 1x - 1,,khi,,x > 1end{array} ight.)

(Rightarrow int_0^2 {left| {1 - x} ight|} dx = int_0^1 {left| {1 - x} ight|} dx + int_1^2 {left| {1 - x} ight|} dx)

(=   int_0^1 {(1 - x)} dx + int_1^2 {(x - 1)} dx)

( = left. {left( {x - frac{{{x^2}}}{2}} ight)} ight|_0^1 + left. {left( {frac{{{x^2}}}{2} - x} ight)} ight|_1^2 = frac{1}{2} + frac{1}{2} = 1)

(egin{array}{l}b),,intlimits_0^{frac{pi }{2}} {{{sin }^2}xdx} = frac{1}{2}intlimits_0^{frac{pi }{2}} {left( {1 - cos 2x} ight)dx} = frac{1}{2}left. {left( {x - frac{{sin 2x}}{2}} ight)} ight|_0^{frac{pi }{2}}= frac{1}{2}.frac{pi }{2} = frac{pi }{4}end{array})

(egin{array}{l}c),,intlimits_0^{ln 2} {frac{{{e^{2x + 1}} + 1}}{{{e^x}}}dx} = intlimits_0^{ln 2} {left( {{e^{2x + 1 - x}} + {e^{ - x}}} ight)dx} = intlimits_0^{ln 2} {left( {{e^{x + 1}} + {e^{ - x}}} ight)dx} = left. {left( {{e^{x + 1}} - {e^{ - x}}} ight)} ight|_0^{ln 2}= {e^{ln 2 + 1}} - {e^{ - ln 2}} - left( {e - 1} ight)= {e^{ln 2}}.e - frac{1}{2} - e + 1= e + frac{1}{2}end{array})

(egin{array}{l}d),,sin 2xcos 2x = sin 2xfrac{{1 + cos 2x}}{2},,, = frac{1}{2}sin 2x + frac{1}{2}sin 2xcos 2x = frac{1}{2}sin 2x + frac{1}{4}sin 4xRightarrow intlimits_0^pi {sin 2xcos 2xdx} = intlimits_0^pi {left( {frac{1}{2}sin 2x + frac{1}{4}sin 4x} ight)dx} = left. {left( { - frac{1}{4}cos 2x - frac{1}{{16}}cos 4x} ight)} ight|_0^pi = - frac{1}{4} - frac{1}{{16}} - left( { - frac{1}{4} - frac{1}{{16}}} ight) = 0end{array}).

soanbailop6.com