Q. 29 - 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.

$\int \left(3.2{\left(1.43\right)}^x-50\right)dx$

Step-by-Step Solution

Answer

The solution of the integral is $3.2\left(\frac{1.{43}^x}{\ln \left(1.43\right)}\right)-50x+C.$

Step 1. Given Information.

The given integral is $\int \left(3.2{\left(1.43\right)}^x-50\right)dx.$

Step 2. Solve.

By solving the integral we get,

$$\begin{gathered} \int \left(3.2{\left(1.43\right)}^x-50\right)dx \\ =3.2\int {\left(1.43\right)}^x dx-\int 50 dx \\ \mathrm{Use} \int a^x=\frac{a^x}{\ln \left(a\right)} \\ =3.2\left(\frac{1.{43}^x}{\ln \left(1.43\right)}\right)-50x+C \end{gathered}$$

Step 3. Verification.

To verify the answer we differentiate $3.2\left(\frac{1.{43}^x}{\ln \left(1.43\right)}\right)-50x+C$ it.

On differentiating we get,

$$\begin{gathered} 3.2\left(\frac{1.{43}^x}{\ln \left(1.43\right)}\right)-50x+C \\ =\frac{3.2}{\ln (1.43)}{\left(1.43\right)}^x-50x+C \\ =8.94{\left(1.43\right)}^x-50x+C \\ =\frac{d}{dx}8.94{\left(1.43\right)}^x-\frac{d}{dx}\left(50x\right)+\frac{d}{dx}\left(C\right) \\ =3.2{\left(1.43\right)}^x-50 \end{gathered}$$

Hence proved.