Calculus Examples

$\frac{d z}{d x} = x^{2} + \cos (x)$

Step 1

Rewrite the equation.

$d z = (x^{2} + \cos (x)) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d z = \int x^{2} + \cos (x) d x$

Step 2.2

Apply the constant rule.

$z + C_{1} = \int x^{2} + \cos (x) d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$z + C_{1} = \int x^{2} d x + \int \cos (x) d x$

Step 2.3.2

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$z + C_{1} = \frac{1}{3} x^{3} + C_{2} + \int \cos (x) d x$

Step 2.3.3

The integral of $\cos (x)$ with respect to $x$ is $\sin (x)$ .

$z + C_{1} = \frac{1}{3} x^{3} + C_{2} + \sin (x) + C_{3}$

Step 2.3.4

Simplify.

$z + C_{1} = \frac{1}{3} x^{3} + \sin (x) + C_{4}$

Step 2.4

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

$z = \frac{1}{3} x^{3} + \sin (x) + K$