Resolved: "Evaluate the integrals ∫0π4cos2(4t−π4)dt"

allhvasstH

Answered question

2021-10-10

Evaluate the integrals

\int_{0}^{\frac{π}{4}} \cos^{2} (4 t - \frac{π}{4}) d t

ottcomn

Skilled2021-10-11Added 97 answers

Given integral is

\int_{0}^{\frac{π}{4}} \cos^{2} (4 t - \frac{π}{4}) d t

Consider

u = 4 t - \frac{π}{4}

Consider

u = 4 t - \frac{π}{4}

Then,

\frac{d u}{d t} = 4

d t = \frac{d u}{4}

Substitute

u = 4 t - \frac{π}{4}

and

d t = \frac{d u}{4}

\int_{0}^{\frac{π}{4}} \cos^{2} (4 t - \frac{π}{4}) d t

= \frac{1}{8} {[u + \frac{\sin (2 u)}{2}]}_{- \frac{π}{4}}^{\frac{3 π}{4}}

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

= \frac{π}{8}

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