Q. 23 - Calculus | StudyQuestionHub

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Q. 23

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 .

$\int (x^{2} - 1) (3 x + 5) d x$

Step-by-Step Solution

Verified

Answer

The answer is $\frac{3 x^{4}}{4} + \frac{5 x^{3}}{3} - \frac{3 x^{2}}{2} - 5 x + C .$

1Step 1. Given Information.

The given integral is $\int (x^{2} - 1) (3 x + 5) d x$ .

2Step 2. Integration.

On integrating,

$\int (x^{2} - 1) (3 x + 5) d x = \int (3 x^{3} + 5 x^{2} - 3 x - 5) d x = 3 \int x^{3} d x + 5 \int x^{2} d x - 3 \int x d x - \int 5 d x = 3 (\frac{x^{4}}{4}) + 5 (\frac{x^{3}}{3}) - 3 (\frac{x^{2}}{2}) - 5 x + C = \frac{3 x^{4}}{4} + \frac{5 x^{3}}{3} - \frac{3 x^{2}}{2} - 5 x + C$

3Step 3. Verification.

On differentiating $\frac{3 x^{4}}{4} + \frac{5 x^{3}}{3} - \frac{3 x^{2}}{2} - 5 x + C,$

$\frac{12 x^{3}}{4} + \frac{15 x^{2}}{3} - \frac{6 x}{2} - 5 = 4 x^{3} + 5 x^{2} - 3 x - 5 = (x^{2} - 1) (3 x + 5)$

Hence Proved.

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