How to integrate sin x?

We will use integration by substitution to integrate sin x.

Answer: The final integral of sin x is − cos x + C

Go through the explanation to understand better.

Explanation:

To solve ∫ sin x dx, let sin x = u ⇒ cos x dx = du

By trigonomteric identity: cos²x = 1 - sin²x, we get

⇒ dx = du / cos x = du / √(1 − u²)

∫sin x dx = ∫u (du / √(1 − u²)) 

= ∫u (1 − u²) ^-1/2 du

= −1/2 ∫(1−u²)^-1/2. (−2u) du

= −1/2 √(1 − u²)/(1/2) + C

= −√(1 − u²) + C

= −√(1 − sin² x) + C

= −√cos² x + C 

= − cos x + C

Thus, the final integral of sin x is − cos x + C.