What is the integral of pi*cos(pix) ?

The integral of pi*cos(pix) is sin(pix)+C

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integral of pi*cos(pix)

\int π \cdot cos (π x) d x

sin (π x) + C

Solution steps

\int π cos (π x) d x

Take the constant out:

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

= π \cdot \int cos (π x) d x

Apply u-substitution

: \int cos (u) \frac{1}{π} d u

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

Take the constant out:

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

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

Use the common integral:

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

= π \frac{1}{π} sin (u)

Substitute back

u = π x

= π \frac{1}{π} sin (π x)

Simplify

π \frac{1}{π} sin (π x) : sin (π x)

= sin (π x)

Add a constant to the solution

= sin (π x) + C

\frac{d}{d x} (5 + x sec (x))

x \to 2 lim (5 x - 3)

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