What is the integral of cos(4x) ?

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

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integral of cos(4x)

\int cos (4 x) d x

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

Solution steps

\int cos (4 x) d x

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 x

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

Add a constant to the solution

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

\int \frac{( x + 1 )}{x ^{2} - 4 x + 12} d x

f (x) = cos (3 x)

\frac{d}{d x} (3 + 2 sin (x))

\int_{0}^{x} a^{2} - x^{2} d x

\frac{d}{d x} (y^{2} - x^{3} + 8)