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.$\int \left(x^2-3x^5-7\right)dx$

Short answer

The answer is$\frac{1}{3}{x}^3-\left(\frac{1}{2}\right)x^6-7x+C.$

Step-by-step solution

Step 1. Given information.

The given integral is$\int \left(x^2-3x^5-7\right)dx.$

Step 2. Integration.

On integrating,

$$\begin{gathered} \int \left(x^2-3x^5-7\right)dx \\ =\int x^{2}dx-\int 3x^{5}dx-\int 7dx \\ =\frac{1}{3}{x}^3-3\left(\frac{1}{6}\right)x^6-7x+C \\ =\frac{1}{3}{x}^3-\left(\frac{1}{2}\right)x^6-7x+C \end{gathered}$$

Step 3. Verification.

On Differentiating$\frac{1}{3}{x}^3-\left(\frac{1}{2}\right)x^6-7x+C$, we get,

$$\begin{gathered} \frac{1}{3}\times 3x^2-\frac{1}{2}\times 6x^5-7 \\ =x^2-3x^5-7 \end{gathered}$$

Hence proved.