Integral of sinxcosx | How to Integrate sinxcosx

The integral of sinxcosx is equal to -(cos2x)/4+C or, (sin²x)/2+C where C is an integration constant. In this post, we will learn how to integrate sinx cosx.

The integration formula of sinxcosx is give as follows:

• ∫sinx cosx dx = -(cos2x)/4+C
• ∫sinx cosx dx = (sin²x)/2+C.
• ∫sinx cosx dx = -(cos²x)/2+C.

Find the integration of sinx cosx

Answer: The integration of sinx cosx is (sin²x)/2+C.

proof:

There are three methods to find the integration of sinx cosx which we will discuss now. They are the substitutions of sinx, cosx and the formula of sin2x.

Integration of Sinx Cosx by Substituting Sinx

The given integral is ∫sinx cosx dx

Put sinx = z.

So we have cosx dx = dz. Now, substituting these values in the given integral we obtain that

∫sinx cosx dx

= ∫z dz

= z²/2+C by the power rule of integration ∫xⁿdx = xⁿ⁺¹/(n+1).

= sin²x/2+C

So the integration of sinxcosx is (sin²x)/2+C where C denotes the constant of integrals.

Integration of Sinx Cosx by Substituting Cosx

∫sinx cosx dx

Put cosx = t.

So -sinx dx = dt. Therefore, we obtain that

∫sinx cosx dx = -∫z dz

= -z²/2+C by the rule ∫xⁿdx = xⁿ⁺¹/(n+1).

= -cos²x/2+C

So the integration of sinxcosx is -(cos²x)/2+C.

ALSO READ:

Integration of xn	Integration of tanx
Integration of secx	Integration of cotx

By the trigonometric formula sin2x = 2sinx cosx, we can write the function sinxcosx as follows:

sinxcosx = (sin2x)/2.

Now, integrating we will get that

∫sinx cosx dx = ∫(sin2x)/2 dx

= 1/2 × ∫sin2x dx

= -1/2 × (cos2x)/2 + C as we know that ∫sin2x dx = -(cosmx)/m.

= -(cos2x)/4+C.

So the integral of sinx cosx is equal to -(cos2x)/4+C.

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