Q. 16 - Calculus

Question

For each pair of functions $u\left(x\right)$ and $v\left(x\right)$ in Exercises 16-18, fill in the blanks to complete each of the following:

(a) $\frac{d}{dx}\left(u\left(x\right)v\left(x\right)\right)$=______

(b) $\int \_\_\_\_dx=u\left(x\right)v\left(x\right)+C$

(c) $\int udv$=______

$u\left(x\right)=x$, $v\left(x\right)=\cos 2x$

Step-by-Step Solution

Answer

The blank is 

Part (a)$-2x\sin 2x+\cos 2x$

Part (b)$-2x\sin 2x+\cos 2x$

Part (c)$-4x\cos 2x+2\sin 2x$

Part (a). Step 1. Given information

The given functions are $u\left(x\right)=x$ and $v\left(x\right)=\cos 2x$.

Part (a). Step 2. Evaluate d d x u x v x .

Differentiate the product of the given functions with respect to x by using integration by parts.

$\begin{array}{rcl}\frac{d}{dx}\left(u\left(x\right)v\left(x\right)\right)& =& \frac{d}{dx}\left(x\cos 2x\right)\\ & =& x\frac{d}{dx}\cos 2x+\cos 2x\frac{d}{dx}x\\ & =& x\left(-2\sin 2x\right)+\cos 2x\\ & =& -2x\sin 2x+\cos 2x\\ & & \end{array}$

Part (b). Step 1. Integration

Integrate the obtained expression for $\frac{d}{dx}\left(u\left(x\right)v\left(x\right)\right)$ with respect to x.

$\begin{array}{rcl}\int \frac{d}{dx}\left(u\left(x\right)v\left(x\right)\right)dx& =& \int \left(-2x\sin 2x+\cos 2x\right)dx\\ u\left(x\right)v\left(x\right)+C& =& \int \left(-2x\sin 2x+\cos 2x\right)dx\end{array}$

From the obtained equation, the integrand missing in part (b) is $-2x\sin 2x+\cos 2x$.

Part (c). Step 1. Differentiation

Differentiate $v\left(x\right)=\cos 2x$ with respect to x to obtain the value of dv.

$\begin{array}{rcl}\frac{dv}{dx}& =& \frac{d}{dx}\cos 2x\\ & =& -2\sin 2x\\ dv& =& -2\sin 2xdx\end{array}$

Part (c). Step 2. Integration

Substitute the given and obtained values to evaluate $\int udv$.

$\begin{array}{rcl}\int udv& =& \int -2x\sin 2xdx\\ & =& -2\int x\sin 2xdx\\ & =& -2\left[x\left(2\cos 2x\right)-\int \left(1\right)\left(2\cos 2x\right)dx\right]\\ & =& -2\left[2x\cos 2x-2\int \cos 2xdx\right]\\ & =& -2\left[2x\cos 2x-2\frac{\sin 2x}{2}\right]\\ & =& -4x\cos 2x+2\sin 2x\end{array}$