Evaluate the integrals

${\int}_{0}^{\frac{\pi}{4}}{\mathrm{cos}}^{2}(4t-\frac{\pi}{4})dt$

allhvasstH
2021-10-10
Answered

Evaluate the integrals

${\int}_{0}^{\frac{\pi}{4}}{\mathrm{cos}}^{2}(4t-\frac{\pi}{4})dt$

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ottcomn

Answered 2021-10-11
Author has **97** answers

Given integral is ${\int}_{0}^{\frac{\pi}{4}}{\mathrm{cos}}^{2}(4t-\frac{\pi}{4})dt$

Consider$u=4t-\frac{\pi}{4}$

Consider$u=4t-\frac{\pi}{4}$

Then,

$\frac{du}{dt}=4$

$dt=\frac{du}{4}$

Substitute$u=4t-\frac{\pi}{4}$ and $dt=\frac{du}{4}$ in ${\int}_{0}^{\frac{\pi}{4}}{\mathrm{cos}}^{2}(4t-\frac{\pi}{4})dt$

$=\frac{1}{8}{[u+\frac{\mathrm{sin}\left(2u\right)}{2}]}_{-\frac{\pi}{4}}^{\frac{3\pi}{4}}$

$=\frac{1}{8}[\frac{3\pi}{4}+\frac{\pi}{4}+\frac{1}{2}(\mathrm{sin}\frac{3\pi}{2}-\mathrm{sin}(-\frac{\pi}{2}))]$

$=\frac{\pi}{8}$

Consider

Consider

Then,

Substitute

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