What is the integral of-cos(5x) ?

The integral of-cos(5x) is -1/5 sin(5x)+C

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integral of-cos(5x)

\int - cos (5 x) d x

- \frac{1}{5} sin (5 x) + C

Solution steps

\int - cos (5 x) d x

Take the constant out:

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

= - \int cos (5 x) d x

Apply u-substitution

: \int cos (u) \frac{1}{5} d u

= - \int cos (u) \frac{1}{5} d u

Take the constant out:

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

= - \frac{1}{5} \cdot \int cos (u) d u

Use the common integral:

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

= - \frac{1}{5} sin (u)

Substitute back

u = 5 x

= - \frac{1}{5} sin (5 x)

Add a constant to the solution

= - \frac{1}{5} sin (5 x) + C

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\frac{d ^{2}}{d x ^{2}} (x^{4})

(x + 2 y) d y = d x

\frac{d}{d x} (x \cdot e^{\frac{1}{x}})