Q. 47 - Calculus | StudyQuestionHub

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Q. 47

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.

$\int \frac{6 x}{3 x^{2} + 1} d x$

Step-by-Step Solution

Verified

Answer

The solution of the integral is $\ln |3 x^{2} + 1| + C .$

1Step 1. Given Information.

The given integral is $\int \frac{6 x}{3 x^{2} + 1} d x .$

2Step 2. Solve.

By solving the integral we get,

$\int \frac{6 x}{3 x^{2} + 1} d x = 6 \int \frac{x}{3 x^{2} + 1} d x Let u = 3 x^{2} + 1, d u = 6 x d x = 6 \int (\frac{1}{6}) \frac{d u}{u} = \int \frac{d u}{u} = \ln |u| + C Substitute back u = 3 x^{2} + 1 = \ln |3 x^{2} + 1| + C$

3Step 3. Verification.

To verify the answer we differentiate $\ln |3 x^{2} + 1| + C$ it.

On differentiating we get,

$\ln |3 x^{2} + 1| + C = \frac{d}{d x} \ln |3 x^{2} + 1| + \frac{d}{d x} C = \frac{6 x}{3 x^{2} + 1}$

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