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in your first question on finding the maximum value of “e” to the power of some function,that function is unclear please post it clearly. and the second goes this way,the question says that integrate that function whose value is maximum,so, $\int_{0}^{\pi }max(cosx,sinx)=\int_{0}^{\frac{\pi }{4}}cosx+\int_{\frac{\pi }{4}}^{\pi }sinx$now in the third question,$\sum_{1}^{100}\frac{1}{4n^{2}-1}=\sum_{1}^{100}\frac{1}{(2n-1)(2n+1)}=\frac{1}{2}(\sum_{1}^{100}\frac{1}{2n-1}-\sum_{1}^{100}\frac{1}{2n+1})=\frac{1}{2}(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}........\frac{1}{199}-\frac{1}{201})=\frac{1}{2}(1-\frac{1}{201})=\frac{100}{201}$