Calculus Examples

$\frac{d y}{d x} = \tan (x) + \sec (x)$

Step 1

Rewrite the equation.

$d y = (\tan (x) + \sec (x)) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \tan (x) + \sec (x) d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int \tan (x) + \sec (x) d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int \tan (x) d x + \int \sec (x) d x$

Step 2.3.2

The integral of $\tan (x)$ with respect to $x$ is $\ln (| \sec (x) |)$ .

$y + C_{1} = \ln (| \sec (x) |) + C_{2} + \int \sec (x) d x$

Step 2.3.3

The integral of $\sec (x)$ with respect to $x$ is $\ln (| \sec (x) + \tan (x) |)$ .

$y + C_{1} = \ln (| \sec (x) |) + C_{2} + \ln (| \sec (x) + \tan (x) |) + C_{3}$

Step 2.3.4

Simplify.

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Step 2.3.4.1

Simplify.

$y + C_{1} = \ln (| \sec (x) |) + \ln (| \sec (x) + \tan (x) |) + C_{4}$

Step 2.3.4.2

Simplify.

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Step 2.3.4.2.1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$y + C_{1} = \ln (| \sec (x) | | \sec (x) + \tan (x) |) + C_{4}$

Step 2.3.4.2.2

To multiply absolute values, multiply the terms inside each absolute value.

$y + C_{1} = \ln (| \sec (x) (\sec (x) + \tan (x)) |) + C_{4}$

Step 2.4

Group the constant of integration on the right side as $C$ .

$y = \ln (| \sec (x) (\sec (x) + \tan (x)) |) + C$