What is the integral of cos(8t) ?

The integral of cos(8t) is 1/8 sin(8t)+C

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integral of cos(8t)

\int cos (8 t) d t

\frac{1}{8} sin (8 t) + C

Solution steps

\int cos (8 t) d t

Apply u-substitution

: \int cos (u) \frac{1}{8} d u

= \int cos (u) \frac{1}{8} d u

Take the constant out:

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

= \frac{1}{8} \cdot \int cos (u) d u

Use the common integral:

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

= \frac{1}{8} sin (u)

Substitute back

u = 8 t

= \frac{1}{8} sin (8 t)

Add a constant to the solution

= \frac{1}{8} sin (8 t) + C

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y^{^{'}} = 3 x + y, y (0) = 7

x \to 0 + lim (x^{2} csc^{2} (x))

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\frac{d}{d x} (\frac{x ^{4} - 5 x ^{3} + x}{x ^{2}})