Calculus Examples

$\frac{d s}{d t} = \frac{1}{t^{2}} + \sin (t)$

Step 1

Rewrite the equation.

$d s = (\frac{1}{t^{2}} + \sin (t)) d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d s = \int \frac{1}{t^{2}} + \sin (t) d t$

Step 2.2

Apply the constant rule.

$s + C_{1} = \int \frac{1}{t^{2}} + \sin (t) d t$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$s + C_{1} = \int \frac{1}{t^{2}} d t + \int \sin (t) d t$

Step 2.3.2

Apply basic rules of exponents.

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Step 2.3.2.1

Move $t^{2}$ out of the denominator by raising it to the $- 1$ power.

$s + C_{1} = \int {(t^{2})}^{- 1} d t + \int \sin (t) d t$

Step 2.3.2.2

Multiply the exponents in ${(t^{2})}^{- 1}$ .

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Step 2.3.2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$s + C_{1} = \int t^{2 \cdot - 1} d t + \int \sin (t) d t$

Step 2.3.2.2.2

Multiply $2$ by $- 1$ .

$s + C_{1} = \int t^{- 2} d t + \int \sin (t) d t$

Step 2.3.3

By the Power Rule, the integral of $t^{- 2}$ with respect to $t$ is $- t^{- 1}$ .

$s + C_{1} = - t^{- 1} + C_{2} + \int \sin (t) d t$

Step 2.3.4

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

$s + C_{1} = - t^{- 1} + C_{2} - \cos (t) + C_{3}$

Step 2.3.5

Simplify.

$s + C_{1} = - \frac{1}{t} - \cos (t) + C_{4}$

Step 2.4

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

$s = - \frac{1}{t} - \cos (t) + K$