Expert Answer to "Evaluate the integral. ∫cos⁡(πx)dx"

waigaK

Answered question

2021-11-09

Evaluate the integral.

\int \cos (π x) d x

irwchh

Skilled2021-11-10Added 102 answers

Step 1 : Analysis
Given :

\int \cos (π x) d x

Step 2 : Simplification
Let the integral be

I = \int \cos (π x) d x

Substituting

u = π x

in the integral I.

\frac{d u}{d x} = \frac{d}{d x} (π x)

\frac{d u}{d x} = π \frac{d}{d x} (x)

\frac{d u}{d x} = π \times 1

\frac{d u}{π} = d x

Step 3 : Solution
The integral I will reduces to

I = \int \cos (u) \frac{d u}{π}

I = \frac{1}{π} \int \cos (u) d u

I = \frac{1}{π} \sin (u) + C

Substituting

u = π x

in the integral I,

\int \cos (π x) d x = \frac{1}{π} \sin (π x) + C

where C is the integral constant.

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