Q. 44 - Calculus

Question

Use integration formulas to solve each integral in Exercises 21–62. You may have to use algebra, educated guess-and-check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating.  

$\int 2\sin x \cos x dx$

Step-by-Step Solution

Answer

The solution of the integral is $-{\cos }^{2}x+C.$

Step 1. Given Information.

The given integral is$\int 2\sin x \cos x dx.$

Step 2. Solve.

By solving the integral we get, 

$$\begin{gathered} \int 2 \sin x \cos x dx \\ =\int {\sin }^{2}x dx \left[2 \sin x \cos x={\sin }^{2}x\right] \\ =-{\cos }^{2}x+C \end{gathered}$$

Step 3. Verification.

To verify the answer we differentiate $-{\cos }^{2}x+C$ it.

On differentiating we get,

$$\begin{gathered} -{\cos }^{2}x+C \\ =\frac{d}{dx}\left(-{\cos }^{2}x\right)+\frac{d}{dx}C \\ =-2\cos x(-\sin x)+0 \\ =2 \sin x \cos x \end{gathered}$$

Hence proved.