(d) evaluate the integral\n int \frac{sec(\theta)}{cos(\theta)} mathrm{d}\theta

Explanation:

Step1: Recall the definition of secant

Recall that $\sec(\theta)=\frac{1}{\cos(\theta)}$. So the integral $\int\frac{\sec(\theta)}{\cos(\theta)}d\theta=\int\frac{\frac{1}{\cos(\theta)}}{\cos(\theta)}d\theta$.

Step2: Simplify the integrand

$\int\frac{\frac{1}{\cos(\theta)}}{\cos(\theta)}d\theta=\int\frac{1}{\cos^{2}(\theta)}d\theta$. Since $\frac{1}{\cos^{2}(\theta)}=\sec^{2}(\theta)$, the integral becomes $\int\sec^{2}(\theta)d\theta$.

Step3: Use the integral formula

The antiderivative of $\sec^{2}(\theta)$ is $\tan(\theta)+C$ (where $C$ is the constant of integration). So $\int\sec^{2}(\theta)d\theta=\tan(\theta)+C$.

Answer:

$\tan(\theta)+C$