Chapter 5: Q 18. (page 428)

For each pair of functions u(x) and v(x) in Exercise, fill in the blanks to complete each of the following:
$u (x) = x^{3}, v (x) = e^{3 x}$
(a) localid="1651745548281" $\frac{d}{d x} (u (x) v (x)) =__$
(b) width="178">∫dx=uxvx+C
(c) $\int u d v =_$

Short Answer

Expert verified

(a) $3 x^{3} e^{3 x} + 2 x^{2} e^{3 x}$

(b) $3 x^{3} e^{3 x} + 2 x^{2} e^{3 x}$

Step by step solution

Step 1. Given information

$u (x) = x^{3}, v (x) = e^{3 x}$

Part (a) Step 1. Explanation

$\begin{array}{rcl} \frac{d}{d x} (x^{3} e^{3 x}) & = & x^{3} \frac{d}{d x} (e^{3 x}) + (e^{3 x}) \frac{d}{d x} (x^{3}) \\ = & 3 x^{3} e^{3 x} + 2 x^{2} e^{3 x} \end{array}$

Part (b) Step 1. Explanation

$\int____d x = u (x) v (x) + C$

As the integration of the value is the anti derivative of the $u (x) v (x)$

$\int (3 x^{3} e^{3 x} + 2 x^{2} e^{3 x}) d x = x^{3} e^{3 x} + C$

Part (c) Step 1. Explanation

$\begin{array}{rcl} \int u d v & = & 3 \int x^{3} e^{3 x} d x \\ = & 3 [x^{3} \frac{e^{3 x}}{3} - \int \frac{e^{3 x}}{3} 3 x^{2}] + C \\ = & 3 [x^{3} \frac{e^{3 x}}{3} - \int e^{3 x} x^{2}] + C \\ = & 3 [x^{3} \frac{e^{3 x}}{3} - (x^{2} \frac{e^{3 x}}{3} - \int \frac{e^{3 x}}{3} 2 x)] + C \\ = & (\frac{x^{3}}{3} - \frac{x^{2}}{3} + \frac{2 x}{9} - \frac{2}{27}) e^{3 x} + C \end{array}$

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For each pair of functions u(x) and v(x) in Exercise, fill in the blanks to complete each of the following:
$u (x) = x^{3}, v (x) = e^{3 x}$
(a) localid="1651745548281" $\frac{d}{d x} (u (x) v (x)) =__$
(b) width="178">∫dx=uxvx+C
(c) $\int u d v =_$

Short Answer

Step by step solution

Step 1. Given information

Part (a) Step 1. Explanation

Part (b) Step 1. Explanation

Part (c) Step 1. Explanation

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For each pair of functions u(x) and v(x) in Exercise, fill in the blanks to complete each of the following:ux=x3,vx=e3x(a) localid="1651745548281" ddxuxvx=______(b) width="178">∫____dx=uxvx+C(c)∫udv=_____

Short Answer

Step by step solution

Step 1. Given information

Part (a) Step 1. Explanation

Part (b) Step 1. Explanation

Part (c) Step 1. Explanation

One App. One Place for Learning.

Most popular questions from this chapter

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For each pair of functions u(x) and v(x) in Exercise, fill in the blanks to complete each of the following:
$u (x) = x^{3}, v (x) = e^{3 x}$
(a) localid="1651745548281" $\frac{d}{d x} (u (x) v (x)) =__$
(b) width="178">∫dx=uxvx+C
(c) $\int u d v =_$