What is the integral of-1+2cos^2(2t) ?

The integral of-1+2cos^2(2t) is 1/4 sin(4t)+C

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integral of-1+2cos^2(2t)

\int - 1 + 2 cos^{2} (2 t) d t

\frac{1}{4} sin (4 t) + C

Solution steps

\int - 1 + 2 cos^{2} (2 t) d t

Rewrite using trig identities

= \int cos (4 t) d t

Apply u-substitution

= \int cos (u) \frac{1}{4} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int cos (u) d u

Use the common integral:

\int cos (u) d u = sin (u)

= \frac{1}{4} sin (u)

Substitute back

u = 4 t

= \frac{1}{4} sin (4 t)

Add a constant to the solution

= \frac{1}{4} sin (4 t) + C

What is the integral of-1+2cos^2(2t) ?
The integral of-1+2cos^2(2t) is 1/4 sin(4t)+C