Calculus Examples

$\int \sec (t) (\sec (t) + \tan (t)) d t$

Step 1

Simplify.

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Step 1.1

Apply the distributive property.

$\int \sec (t) \sec (t) + \sec (t) \tan (t) d t$

Step 1.2

Multiply $\sec (t) \sec (t)$ .

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Step 1.2.1

Raise $\sec (t)$ to the power of $1$ .

$\int \sec^{1} (t) \sec (t) + \sec (t) \tan (t) d t$

Step 1.2.2

Raise $\sec (t)$ to the power of $1$ .

$\int \sec^{1} (t) \sec^{1} (t) + \sec (t) \tan (t) d t$

Step 1.2.3

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

$\int {\sec (t)}^{1 + 1} + \sec (t) \tan (t) d t$

Step 1.2.4

Add $1$ and $1$ .

$\int \sec^{2} (t) + \sec (t) \tan (t) d t$

Step 2

Split the single integral into multiple integrals.

$\int \sec^{2} (t) d t + \int \sec (t) \tan (t) d t$

Step 3

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

$\tan (t) + C + \int \sec (t) \tan (t) d t$

Step 4

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

$\tan (t) + C + \sec (t) + C$

Step 5

Simplify.

$\tan (t) + \sec (t) + C$