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.

$\int ({\sin }^{2}x+{\cos }^{2}x)dx$

Short answer

The solution of the integral is $x+C.$

Step-by-step solution

Step 1. Given Information. 

The given integral is$\int ({\sin }^{2}x+{\cos }^{2}x)dx.$

Step 2. Solve. 

By solving the integral we get,

$$\begin{gathered} \int ({\sin }^{2}x+{\cos }^{2}x)dx \\ =\int 1dx\left[{\sin }^{2}x+{\cos }^{2}x=1\right] \\ =x+C \end{gathered}$$

Step 3. Verification. 

To verify the answer we differentiate $x+C$it.

On differentiating we get,

$$\begin{gathered} x+C \\ =\frac{d}{dx}x+\frac{d}{dx}C \\ =1+0 \\ =1 \\ ={\sin }^{2}x+{\cos }^{2}x\left[{\sin }^{2}x+{\cos }^{2}x=1\right] \end{gathered}$$

Hence proved.