Calculus Examples

$r (θ) = \sec (θ) \tan (θ) - 4 e^{θ}$

Step 1

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

$R (θ) = \int r (θ) d θ$

Step 2

Set up the integral to solve.

$R (θ) = \int \sec (θ) \tan (θ) - 4 e^{θ} d θ$

Step 3

Split the single integral into multiple integrals.

$\int \sec (θ) \tan (θ) d θ + \int - 4 e^{θ} d θ$

Step 4

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

$\sec (θ) + C + \int - 4 e^{θ} d θ$

Step 5

Since $- 4$ is constant with respect to $θ$ , move $- 4$ out of the integral.

$\sec (θ) + C - 4 \int e^{θ} d θ$

Step 6

The integral of $e^{θ}$ with respect to $θ$ is $e^{θ}$ .

$\sec (θ) + C - 4 (e^{θ} + C)$

Step 7

Simplify.

$\sec (θ) - 4 e^{θ} + C$

Step 8

The answer is the antiderivative of the function $r (θ) = \sec (θ) \tan (θ) - 4 e^{θ}$ .

$R (θ) =$ $\sec (θ) - 4 e^{θ} + C$