# Evaluate the integral. integral fraction {3 sin x + 4 sec x tan x}{cos x} dx A) sec^2x -3 ln|cos...

## Question:

Evaluate the indefinite integral:

{eq}I= \int \frac {3 \sin x + 4 \sec x \tan x}{\cos x }\; dx {/eq}

(a) {eq}2 \sec^2x -3 \ln|\cos x| + C,\text{ or }2 \tan^2 x - 3 \ln|\cos x| + C. {/eq}

(b) {eq}2 \tan^2 x - 3 \ln|\cos x| + C,\text{ or }2 \tan^2x + 3 \ln|\cos x| + C . {/eq}

(c) {eq}4 \sec^2 x + 3 \ln|\cos x| + C,\text{ or }4 \tan^2 x + 3 \ln|\cos x| + C . {/eq}

(d) {eq}\frac {1}{2} \sec^2 x - \ln|\sec x| + C,\text{ or }\frac {1}{2} \tan^2 x - \ln|\sec x| + C. {/eq}

## Indefinite Integrals; Antiderivatives:

We will rewrite the integrand {eq}\frac {3 \sin x + 4 \sec x \tan x}{\cos x } {/eq} in an equivalent by using some basic trigonometric identities. To find an antiderivative we will recall some basic derivatives:

{eq}\begin{align*} \frac{d\left(\ln(u(x))\right)}{dx}&=\frac{1}{u(x)}u'(x),\\ \frac{d\left(\sec x\right)}{dx} &=\sec x \tan x. \end{align*} {/eq}

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The valid answer is option (a) as we'll explain below:

{eq}\begin{align*} I&= \int \frac {3 \sin x + 4 \sec x \tan x}{\cos x }\; dx\\ &= \int... 