QuestionFind the antiderivatives of $\frac{d x}{d t}=2 t^{-1}+5$ . What is $x$ ? Include the constant $C$ .

Studdy Solution

STEP 1

Assumptions1. The given function is $\frac{d x}{d t}= t^{-1}+5$ . We are asked to find the antiderivative of the given function3. The antiderivative of a function is the reverse process of differentiation4. The constant of integration, denoted by $C$ , is added at the end of the antiderivative

STEP 2

The antiderivative of a function is found by reversing the process of differentiation. In this case, we need to find the antiderivative of $2 t^{-1}+5$ .
The general formula for finding the antiderivative is $\int f(x) \, dx = F(x) + C$ where $(x)$ is the antiderivative of $f(x)$ and $C$ is the constant of integration.

STEP 3

We can break down the given function into two parts and find the antiderivative of each part separately. $\int (2 t^{-1}+5) \, dt = \int2 t^{-1} \, dt + \int5 \, dt$

STEP 4

Now, we can find the antiderivative of each part.
The antiderivative of $2 t^{-1}$ is $2 \ln |t|$ and the antiderivative of $$ is $t$.
So, $\int2 t^{-1} \, dt =2 \ln |t|$ $\int \, dt =t$

STEP 5

Now, we can add the antiderivatives of the two parts together and add the constant of integration $C$ .
$x =2 \ln |t| +5t + C$ So, the antiderivative of the given function is $x =2 \ln |t| +5t + C$ .

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