Q. 28 - 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. 

$\int \left(\frac{3}{x^2}-4\right)dx$

Step-by-Step Solution

Answer

The answer is$\frac{-3}{x}-4x+C$.

Step 1. Given Information.

The given integral is $\int \left(\frac{3}{x^2}-4\right)dx.$

Step 2. Integration.

On integrating, we get,

$$\begin{gathered} \int \left(\frac{3}{x^2}-4\right)dx \\ =\int \frac{3}{x^2}dx-\int 4 \\ =3\int \frac{1}{x^2}dx-\int 4 \\ =3\left(\frac{x^{-2+1}}{-2+1}\right)-4x+C \\ =3\left(\frac{x^{-1}}{-1}\right)-4x+C \\ =\frac{-3}{x}-4x+C \end{gathered}$$

Step 3. Verification.

On differentiating $\frac{-3}{x}-4x+C$, we get,

$=\frac{3}{x^2}-4$

Hence proved.