What is the integral of (cos^2(t))/(cos(c^6)t) ?

The integral of (cos^2(t))/(cos(c^6)t) is 1/2 sec(c^6)(ln|t|+\Ci(2t))+C

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integral of (cos^2(t))/(cos(c^6)t)

\int \frac{cos ^{2} ( t )}{cos ( c ^{6} ) t} d t

\frac{1}{2} sec (c^{6}) (ln ∣ t ∣ + Ci (2 t)) + C

Solution steps

\int \frac{cos ^{2} ( t )}{cos ( c ^{6} ) t} d t

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{cos ( c ^{6} )} \cdot \int \frac{cos ^{2} ( t )}{t} d t

Rewrite using trig identities

= \frac{1}{cos ( c ^{6} )} \cdot \int \frac{1 + cos ( 2 t )}{2 t} d t

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{cos ( c ^{6} )} \cdot \frac{1}{2} \cdot \int \frac{1 + cos ( 2 t )}{t} d t

Expand

\frac{1 + cos ( 2 t )}{t} : \frac{1}{t} + \frac{cos ( 2 t )}{t}

= \frac{1}{cos ( c ^{6} )} \cdot \frac{1}{2} \cdot \int \frac{1}{t} + \frac{cos ( 2 t )}{t} d t

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{cos ( c ^{6} )} \cdot \frac{1}{2} (\int \frac{1}{t} d t + \int \frac{cos ( 2 t )}{t} d t)

\int \frac{1}{t} d t = ln ∣ t ∣

\int \frac{cos ( 2 t )}{t} d t = Ci (2 t)

= \frac{1}{cos ( c ^{6} )} \cdot \frac{1}{2} (ln ∣ t ∣ + Ci (2 t))

Simplify

\frac{1}{cos ( c ^{6} )} \cdot \frac{1}{2} (ln ∣ t ∣ + Ci (2 t)) : \frac{1}{2} sec (c^{6}) (ln ∣ t ∣ + Ci (2 t))

= \frac{1}{2} sec (c^{6}) (ln ∣ t ∣ + Ci (2 t))

Add a constant to the solution

= \frac{1}{2} sec (c^{6}) (ln ∣ t ∣ + Ci (2 t)) + C

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What is the integral of (cos^2(t))/(cos(c^6)t) ?
The integral of (cos^2(t))/(cos(c^6)t) is 1/2 sec(c^6)(ln|t|+\Ci(2t))+C