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.
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).
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).
So the integration of sinxcosx is -(cos2x)/2+C.
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.
So the integral of sinx cosx is equal to -(cos2x)/4+C.
Have You Read These Integrations?
Q1: What is the integration of sinxcosx?
Answer: The integration of sinxcosx is given by ∫sinx cosx dx = -(cos2x)/4+C where C is an integral constant.