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Chapter 4: Q. 23 (page 404)

Indefinite integrals: Use integration formulas, algebra, and educated guess-and-check strategies to find the following integrals.
$鈭� (2 x 3^{x} + x^{2} (\ln 3) 3^{x}) d x$

Short Answer

Expert verified

The solution is: $x^{2} 3^{x} + C$

Step by step solution

01

Given

The given integral is: $鈭� (2 x 3^{x} + x^{2} (\ln 3) 3^{x}) d x$

02

To Find

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

03

Calculation

Let,

$u = x^{2} 3^{x} d u = (2 x 3^{x} + x^{2} 3^{x} \ln (3)) d x d x = \frac{d u}{(2 x 3^{x} + x^{2} 3^{x} \ln (3))}$

Using substitution we get:

鈭� (2 x 3^{x} + x^{2} (\ln 3) 3^{x}) d x = 鈭� (2 x 3^{x} + x^{2} (\ln 3) 3^{x}) \frac{d u}{(2 x 3^{x} + x^{2} 3^{x} \ln (3))} = 鈭� d u = u + C = x^{2} 3^{x} + C

Where, C = integrating constant.

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Most popular questions from this chapter

Use the Fundamental Theorem of Calculus to find the exact values of the given definite integrals. Use a graph to check your answer.

${鈭�}_{0}^{1 / 2} 鈥� \frac{1}{1 + 4 x^{2}} d x$

Find the sum or quantity without completely expanding or calculating any sums.

Given ${鈭�}_{k = 1}^{4} a_{k} = 7$ and $, {鈭�}_{k = 0}^{4} b_{k} = 10$ , $a_{0} = 2$ . Find the value of ${鈭�}_{k = 0}^{4} (2 a_{k} + 3 b_{k})$ .

Properties of addition: State the associative law for addition, the commutative law for addition, and the distributive law for multiplication over addition of real numbers. (You may have to think back to a previous algebra course.)

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The absolute area between the graph of f and the x-axis on [a, b] is equal to $| {鈭�}_{a}^{b} f (x) d x |$ .

(b) True or False: The area of the region between f(x) = x 鈭� 4 and g(x) = $- x^{2}$ on the interval [鈭�3, 3] is negative.

(c) True or False: The signed area between the graph of f on [a, b] is always less than or equal to the absolute area on the same interval.

(d) True or False: The area between any two graphs f and g on an interval [a, b] is given by ${鈭�}_{a}^{b} (f (x) - g (x)) d x$ .

(e) True or False: The average value of the function f(x) = $x^{2} - 3$ on [2, 6] is

\frac{f (6) + f (2)}{2}

\frac{33 + 1}{2}

(f) True or False: The average value of the function f(x) = $x^{2} - 3$ on [2, 6] is $\frac{f (6) - f (2)}{4}$ = $\frac{33 - 1}{4}$ = 8.

(g) True or False: The average value of f on [1, 5] is equal to the average of the average value of f on [1, 2] and the average value of f on [2, 5].

(h) True or False: The average value of f on [1, 5] is equal to the average of the average value of f on [1, 3] and the average value of f on [3, 5].

Fill in each of the blanks:

(a) $鈭� x^{6} d x = + C .$

(b) is an antiderivative of role="math" localid="1648619282178" $x^{6} .$

(c) The derivative of is $x^{6} .$

See all solutions

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