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Chapter 4: Q. 21 (page 404)

Indefinite integrals: Use integration formulas, algebra, and educated guess-and-check strategies to find the following integrals.
$\int \frac{3}{1 + 4 x} d x$

Short Answer

Expert verified

The solution is: $\frac{3}{4} \ln |1 + 4 x| + C$

Step by step solution

01

Given

The given integral is: $\int \frac{3}{1 + 4 x} d x$

02

To Find

Use integration formulas, algebra, and educated guess-and-check strategies to find the given integral.

03

Calculation

Let,

$u = 1 + 4 x d u = 4 d x \frac{d u}{4} = d x$

Using substitution we get:

$\int \frac{3}{1 + 4 x} d x = \int \frac{3}{u} \frac{d u}{4} = \frac{3}{4} \int \frac{d u}{u} = \frac{3}{4} \ln |u| + C = \frac{3}{4} \ln |1 + 4 x| + C$

Where, $C$ = integrating constant.

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Most popular questions from 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 as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating .

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

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.

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

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. (Hint for Exercise 54: $\tan (x) = \frac{\sin (x)}{\cos (x)}$ ).

$\int \frac{2 x \ln x - x}{{(\ln x)}^{2}} d x .$

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.

$\int \frac{x + 1}{\sqrt{x}} d x$

Without using absolute values, how many definite integrals would we need in order to calculate the absolute area between f(x) = sin x and the x-axis on $[- \frac{π}{2}, 2 π]$ ?

Will the absolute area be positive or negative, and why? Will the signed area will be positive or negative, and why?

See all solutions

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