Calculus Examples

$\frac{\sin (θ)}{\cos^{2} (θ)}$

Step 1

Write $\frac{\sin (θ)}{\cos^{2} (θ)}$ as a function.

$f (θ) = \frac{\sin (θ)}{\cos^{2} (θ)}$

Step 2

The function $F (θ)$ can be found by finding the indefinite integral of the derivative $f (θ)$ .

$F (θ) = \int f (θ) d θ$

Step 3

Set up the integral to solve.

$F (θ) = \int \frac{\sin (θ)}{\cos^{2} (θ)} d θ$

Step 4

Multiply by $1$ .

$\int \frac{\sin (θ) \cdot 1}{\cos^{2} (θ)} d θ$

Step 5

Factor $\cos (θ)$ out of $\cos^{2} (θ)$ .

$\int \frac{\sin (θ) \cdot 1}{\cos (θ) \cos (θ)} d θ$

Step 6

Separate fractions.

$\int \frac{\sin (θ)}{\cos (θ)} \cdot \frac{1}{\cos (θ)} d θ$

Step 7

Convert from $\frac{\sin (θ)}{\cos (θ)}$ to $\tan (θ)$ .

$\int \tan (θ) \frac{1}{\cos (θ)} d θ$

Step 8

Convert from $\frac{1}{\cos (θ)}$ to $\sec (θ)$ .

$\int \tan (θ) \sec (θ) d θ$

Step 9

Since the derivative of $\sec (θ)$ is $\tan (θ) \sec (θ)$ , the integral of $\tan (θ) \sec (θ)$ is $\sec (θ)$ .

$\sec (θ) + C$

Step 10

The answer is the antiderivative of the function $f (θ) = \frac{\sin (θ)}{\cos^{2} (θ)}$ .

$F (θ) =$ $\sec (θ) + C$