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Evaluate of each of the following int...

Evaluate of each of the following integrals (1-15): `int_(-pi//4)^(pi//4)(x^(11)-3 x^9+5x^7-x^5+1)/(cos^2x)dx`

Text Solution

Verified by Experts

` Let I=int_{-frac{pi}{4}}^{frac{x}{4}} frac{x^{11}-3 x^{9}+5 x^{7}-x^{5}+1}{cos ^{2} x} d x `
` =int_{-frac{pi}{4}}^{frac{pi}{4}} frac{x^{11}-3 x^{9}+5 x^{7}-x^{5}}{cos ^{2} x} d x+int_{-frac{pi}{4}}^{frac{pi}{4}} frac{1}{cos ^{2} x} d x `
` =int_{-frac{pi}{4}}^{frac{pi}{4}} frac{x^{11}-3 x^{9}+5 x^{7}-x^{5}}{cos ^{2} x} d x+int_{-frac{x}{4}}^{frac{x}{4}} sec ^{2} x d x `
` =I_{1}+I_{2}`
Now,
Consider
` f(x)=frac{x^{11}-3 x^{9}+5 x^{7}-x^{5}}{cos ^{2} x}`
` ...
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