What is the integral of cos(4θ) ?

The integral of cos(4θ) is 1/4 sin(4θ)+C

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integral of cos(4θ)

\int cos (4 θ) d θ

\frac{1}{4} sin (4 θ) + C

Solution steps

\int cos (4 θ) d θ

Apply u-substitution

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

= \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 θ

= \frac{1}{4} sin (4 θ)

Add a constant to the solution

= \frac{1}{4} sin (4 θ) + C

\int_{2}^{5} x x - 1 d x

y = \frac{t}{4 t - 5}

y^{^{''}} + y^{^{'}} + y = sin^{2} (x)

x \to 1 - lim (\frac{3}{x ^{2} - 1})

y = - x^{2} + 5 x - 3, y = x^{2} - 3 x + 3