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Evaluate (a) ? ( ? x + e x ? sec x tan x ) d x (b) ? ( sin 2 x + x sin x sin x ) d x (c) ?...

Question:

Evaluate

(a) {eq}\int ( \sqrt{x} + e^x - \sec x \tan x) dx {/eq}

(b) {eq}\int \begin{pmatrix} \sin^2 x + x \sin x \\ \hline \sin x \end{pmatrix} dx {/eq}

(c) {eq}\int \cot^2 x \ dx {/eq}

(d) {eq}\int \sin x \cos x \csc x \cot x \sec x \tan x \ dx {/eq}

Integrals:

The given problem on indefinite integrals is very basic and can be easily solved by applying the basic rules and standard trigonometric formulae of the integral calculus. {eq}\begin{align*} \int e^x \ dx = \frac{e^x}{\frac{d}{dx}(x)} + c \ \ ; \ \ \int (\sec x \tan x) \ dx = \sec x + c \ \ ; \ \ \int \csc^2 x \ dx = -\cot x + c \end{align*} {/eq}.

Answer and Explanation:


(a) {eq}\int ( \sqrt{x} + e^x - \sec x \tan x) dx {/eq}

Integrating the above, we get:

{eq}\begin{align*} \ & = \left(\frac{x^{3/2}}{\frac{3}{2}} + \frac{e^x}{\frac{d}{dx}(x)} - \sec x \right) + c \\ \\ \ & = \frac{2}{3} x^{3/2} + \frac{e^x}{1} - \sec x + c \\ \\ \ & = \frac{2}{3} x^{3/2} + e^x - \sec x + c \end{align*} {/eq}


(b) {eq}\int \begin{pmatrix} \sin^2 x + x \sin x \\ \hline \sin x \end{pmatrix} dx {/eq}

Simplifying and integrating the above, we get:

{eq}\begin{align*} \ & = \int (\sin x+ x) \ dx \\ \\ \ & = -\cos x+ \frac{x^2}{2} + c \end{align*} {/eq}


(c) {eq}\int \cot^2 x \ dx {/eq}

Rewriting and integrating the above, we get:

{eq}\begin{align*} \ & = \int (\csc^2 x- 1) \ dx \ \ \ \ \ \ \ \ \ \ \ \ \ \left[ \ (1+ \cot^2 x) = \csc^2 x \right] \\ \\ \ & = -\cot x- x + c \end{align*} {/eq}


(d) {eq}\int \sin x \cos x \csc x \cot x \sec x \tan x \ dx {/eq}

Simplifying and integrating the above, we get:

{eq}\begin{align*} \ & = \int ( \sin x \cdot \cos x \cdot \frac{1}{\sin x} \cdot \frac{\cos x}{\sin x} \cdot \frac{1}{\cos x} \cdot \frac{\sin x}{\cos x} ) \ dx \\ \\ \ & = \int ( 1 ) \ dx \\ \\ \ & = x + c \end{align*} {/eq}


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Basic Calculus: Rules & Formulas

from Calculus: Tutoring Solution

Chapter 3 / Lesson 6
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