To determine

To evaluate:

∫0π41+cos2⁡θcos2⁡θdθ

Answer

1+π4

Explanation

1) Concept:

By using the fundamental theorem of calculus and the rules of integration, evaluate the given integral.

The fundamental theorem of calculus:

If f is continuous on [a, b], then ∫abfxdx=Fb-F(a) where F is the anti derivative of f.

2) Formula:

∫fx+gxdx=∫fxdx+∫gxdx

∫sec2⁡xdx=tan⁡x+C

sec⁡x=1cos⁡x

3) Given:

∫0π41+cos2⁡θcos2⁡θdθ

4) Calculation:

Consider, ∫0π41+cos2⁡θcos2⁡θdθ

Dividing by the denominator separately to separate the fraction we have,

∫0π41+cos2⁡θcos2⁡θdθ= ∫0π41cos2⁡θ+1dθ= ∫0π4sec2⁡θ+1dθ

 Applying the rules of integral:

After separating the integrals,

 ∫0π4sec2⁡θ+1dθ== ∫0π4sec2⁡θdθ+∫0π41dθ

By applying the fundamental theorem of calculus and the power rule,

=[tan⁡θ+θ]oπ4

=tan⁡π4+π4-tan⁡0+0

=1+π4-0+0

=1+π4

Conclusion:

Therefore,

∫0π41+cos2⁡θcos2⁡θdθ=1+π4