Calculus Examples

$\int \frac{\sec (θ)}{\cos (θ)} d θ$

Step 1

Rewrite $\sec (θ)$ in terms of sines and cosines.

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

Step 2

Rewrite $\frac{\frac{1}{\cos (θ)}}{\cos (θ)}$ as a product.

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

Step 3

Multiply $\frac{1}{\cos (θ)}$ by $\frac{1}{\cos (θ)}$ .

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

Step 4

Simplify the denominator.

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Step 4.1

Raise $\cos (θ)$ to the power of $1$ .

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

Step 4.2

Raise $\cos (θ)$ to the power of $1$ .

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

Step 4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.4

Add $1$ and $1$ .

$\int \frac{1}{\cos^{2} (θ)} d θ$

Step 5

Simplify with factoring out.

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Step 5.1

Rewrite $1$ as $1^{2}$ .

$\int \frac{1^{2}}{\cos^{2} (θ)} d θ$

Step 5.2

Rewrite $\frac{1^{2}}{\cos^{2} (θ)}$ as ${(\frac{1}{\cos (θ)})}^{2}$ .

$\int {(\frac{1}{\cos (θ)})}^{2} d θ$

Step 6

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

$\int \sec^{2} (θ) d θ$

Step 7

Since the derivative of $\tan (θ)$ is $\sec^{2} (θ)$ , the integral of $\sec^{2} (θ)$ is $\tan (θ)$ .

$\tan (θ) + C$