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Find the value of `int_(-2)^(2)(sin^(-1)(sinx)+cos^(-1)(cosx))/((1+x^(2))(1+[(x^(2))/5]))dx`, where [.] represents the greatest integer function. |
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Answer» `I=int_(-2)(2)(sin^(-1)(sinx)+cos^(-1)(cosx))/((1+x^(2))(1+[(x^(2))/5]))` `sin^(-1)(sinx)` is an odd function. Also for `-2lt xlt 2, 0le x^(2)lt4`. `:. [(x^(2))/5]=0` `:.I=int_(-1)^(2)(cos^(-1)(cosx))/(1+x^(2))dx` `=int_(0)^(2)(2cos^(-1)(cosx))/(1+x^(2))dx` `=int_(0)^(2)(2x)/(1+x^(2)dx` `=[log_(e)(1+x^(2))]_(0)^(2)` `=log_(e)4` |
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