Expert answers to "Find integral. ∫cos⁡5xdx"

dailinoyf

Answered question

2021-11-19

Find integral.

\int \cos 5 x d x

Archie Griffin

Beginner2021-11-20Added 9 answers

Step 1
We have to find the integral

I = \int \cos (5 x) d x

Step 2

I = \int \cos (5 x) d x

let 5x=t
5dx=dt

d x = \frac{d t}{5}

\Rightarrow I = \int \cos (t) \frac{d t}{5}

\Rightarrow I = \frac{1}{5} \int \cos (t) d t

\Rightarrow I = \frac{1}{5} (\sin t) + C

\Rightarrow I = \frac{1}{5} \sin t + C

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

(answer)

Elizabeth Witte

Beginner2021-11-21Added 24 answers

Step 1: Use Integration by Substitution.
Let u=5x, du=5 dx, then

d x = \frac{1}{5} d u

Step 2: Using u and du above, rewrite

\int \cos 5 x d x

\int \frac{\cos u}{5} d u

Step 3: Use Constant Factor Rule:

\int c f (x) d x = c \int f (x) d x

\frac{1}{5} \int \cos u d u

Step 4: Use Trigonometric Integration: the integral of

\cos u i s \sin u

\frac{\sin u}{5}

Step 5: Substitute u=5x back into the original integral.

\frac{\sin 5 x}{5}

Step 6: Add constant.

\frac{\sin 5 x}{5} + C

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