Q. 45 - Calculus | StudyQuestionHub

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Q. 45

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.

$\int (x^{2} e^{x} + 2 x e^{x}) d x$

Step-by-Step Solution

Verified

Answer

The solution of the integral is $e^{x} x^{2} + C .$

1Step 1. Given Information.

The given integral is $\int (x^{2} e^{x} + 2 x e^{x}) d x .$

2Step 2. Solve.

By solving the integral we get,

$\int (x^{2} e^{x} + 2 x e^{x}) d x = \int e^{x} x (x + 2) d x = (x + 2) (e^{x} x - e^{x}) - \int e^{x} x - e^{x} d x = (x + 2) (e^{x} x - e^{x}) - (e^{x} x - 2 e^{x}) = e^{x} x^{2} + C$

3Step 3. Verification.

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

On differentiating we get,

$e^{x} x^{2} + C = \frac{d}{d x} e^{x} x^{2} + \frac{d}{d x} C = x^{2} e^{x} + 2 x e^{x} + 0 = x^{2} e^{x} + 2 x e^{x}$

Hence proved.

Other exercises in this chapter

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 a

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 a

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 a

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 a