Integral of sinx cosx

Integration of sin x cos x dx

Integration of sinx cosx dx can be find easily using different methods but before that we will recall some of trigonometric identities like sin 2x = 2 sin x cos x. In this blog we will find the Integral of sinx cosx using substitution method and formula of sin 2x.

As you know, Integral of sinx cosx can be written as ∫sin x cos x dx = (-1/4) cos 2x + C in mathematical terms, Where C is a integral constant, dx means integration with respect x and ∫ denotes integration sign.

Integration of Sin x Cos x Formula

Now lets start to write formula of Integral of sinx cosx using different approaches. We will see the derivation for the Integration of Sin x Cos x Formula using sin x formula.

• ∫ sin x cos x dx = (-1/4) cos 2x + C [When evaluated using the sin 2x formula]
• ∫ sin x cos x dx = (-1/2) cos²x + C [When evaluated by substituting cos x]
• ∫ sin x cos x dx = (1/2) sin²x + C [When evaluated by substituting sin x]

Derivation of Integral of Sin x Cos x Using Sin 2x Formula

As we motioned this formula for integration of sinx cosx can be derived using sin2x formula

• sin 2x = 2 sin x cos x
• ∫sin (ax) dx = (-1/a) cos (ax) + C

Using above two formula sin 2x and ∫sin (ax) dx. We have ∫ sin x cos x dx = ∫(2/2) sin x cos x dx [Multiplying and dividing sin x cos x by 2].

• ∫ sin x cos x dx = (1/2) ∫2 sin x cos x dx
• ∫ sin x cos x dx = (1/2) ∫sin 2x dx [Using sin 2x = 2 sin x cos x]
• ∫ sin x cos x dx = (1/2) (-1/2) cos 2x + C
• ∫ sin x cos x dx = (-1/4) cos 2x + C

Hence we have derived the integral of sin x cos x using the sin 2x formula. Please follow this link of Explanation of Mathematics formula – Click