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Let int_(0)^(oo) (sinx)/x dx=alpha Then match the following lists and choose the correct code. : |
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Answer» Let `5x=t` `:. dx=1/5dt` `:. I=int_(0)^(oo) (sin(t))/(t//5) . (dt)/5=int_(0)^(oo) (sint)/t dt=alpha` b. `I=int_(0)^(sin^(2)x)dx` `=int_(0)^(oo) (sin^(2)x)(1/(x^(2)))dx` `=[sin^(2)x((-1)/x)]_(0)^(oo) -int_(0)^(oo) (2sinxcosx)xx((-1)/x)dx` `=[(-sin^(2)x)/x]_(0)^(oo) +int_(0)^(oo) (sin2x)/x dx` put `2x=t` `=lim_(XTO oo) [-(sin^(2)x)/x]-lim_(xto oo) [-(sin^(2)x)/x]+alpha` `=0-0+alpha=alpha` c. `I=int_(0)^(oo) (sin^(3)x)/x dx` `I=1/4[3int_(0)^(oo) (sinx)/x dx-int_(0)^(oo) (sin3x)/x dx]` `=1/4[3alpha-alpha]=(alpha)/2` d. `int_(0)^(oo) (sink_(1)xcosk_(2)x)/xdx` `=1/2int_(0)^(oo) (2sink_(1)xcosk_(2)x)/xdx` `=1/2int_(0)^(oo) (sin(k_(1)+k_(2))x+sin(k_(1)-k_(2))x)/x dx` `=1/2 (alpha+alpha)=alpha` `:. int_(0)^(oo) (sin(k_(1)x).cosk_(2)x)/x dx-alpha=0` |
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