Evaluate the following definite integral `int_0^(pi/2) cos^4 x dx`

Explanation

`int_{0}^{pi / 2} cos ^{4} x d x=int_{0}^{pi / 2} cos ^{2} x x cos ^{2} x d x`
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Rightarrow int_{0}^{frac{pi}{2}} cos ^{4} x d x=int_{0}^{frac{pi}{2}}(frac{1+cos 2 x}{2}) times(frac{1+cos 2 x}{2}) d x`
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(because 1+cos 2 theta=2 cos ^{2} theta) `
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Rightarrow int_{0}^{frac{pi}{2}} cos ^{4} x d x=(frac{1}{4}) int_{0}^{frac{pi}{2}}(1+cos 2 x)(1+cos 2 x) d x`
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Rightarrow int_{0}^{frac{pi}{2}} cos ^{4} x d x=(frac{1}{4}) int_{0}^{frac{pi}{2}}(1+cos ^{2} 2 x+2 cos 2 x) d x`
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Rightarrow int_{0}^{frac{pi}{2}} cos ^{4} x d x=(frac{1}{4}) int_{0}^{frac{pi}{2}}(1+frac{(1+cos 4 x)}{2}+2 cos 2 x) d x`
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Rightarrow int_{0}^{frac{pi}{2}} cos ^{4} x d x=(frac{1}{8}) int_{0}^{frac{pi}{2}}(2+(1+cos 4 x)+4 cos 2 x) d x`
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Rightarrow int_{0}^{frac{pi}{2}} cos ^{4} x d x=(frac{1}{8})[int_{a}^{frac{pi}{2}}(2 d x+int_{0}^{frac{pi}{2}}(1+cos 4 x) d x+int_{0}^{frac{pi}{2}}(4 cos 2 x) d x]. `
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