DE-BOOK Antiderivatives

Section 3.1 Antiderivatives

When you find an antiderivative, you’re actually solving a basic differential equation. For example, consider finding the general antiderivative of

x^{2} .

The calculus approach would compute the integral

\int x^{2} d x = \frac{1}{3} x^{3} + c .

The differential equations approach, on the other hand, seeks all functions whose derivative is

x^{2} .

This means solving any of the following equivalent equations for

y :

\begin{matrix} (12) & \frac{d}{d x} [y] = x^{2}, \frac{d y}{d x} = x^{2}, or y^{'} = x^{2} \end{matrix}

To solve for

y,

we integrate both sides with respect to

x,

like so:

\begin{aligned} \int \frac{d}{d x} [y] d x & = \int x^{2} d x \\ y + c_{1} & = \frac{1}{3} x^{3} + c_{2} \\ y & = \frac{1}{3} x^{3} + c_{2} - c_{1} ⟹ y = \frac{1}{3} x^{3} + c \end{aligned}

where

c = c_{2} - c_{1}

is a constant. Although this method might seem excessive for simple problems, it illustrates the core concept of isolating the dependent variable and expressing it in terms of the independent variable,

x .

Note 1. Combining Constants is very common in differential equations.

Example 2.

2 y^{'} - 4 \sin x = 2, y (0) = 5 .

Solution. Solution

Start by isolating the derivative,

y^{'},

on one side of the equation

\begin{aligned} y^{'} & = 1 + 2 \sin x \end{aligned}

Integrate both sides with respect to

x

to leave us with

y

on the left side

\begin{aligned} \int y^{'} d x & = \int (1 + 2 \sin x) d x \\ y + c_{1} & = x - 2 \cos x + c_{2} \\ y & = x - 2 \cos x + c_{2} - c_{1} \leftarrow c \\ y (x) & = x - 2 \cos x + c \end{aligned}

Finally, apply the initial condition

y (0) = 5

to find

c

\begin{aligned} y (0) & = 5 \\ (0) - 2 \cos (0) + c & = 5 \\ 0 - 2 + c & = 5 \\ c & = 7 \end{aligned}

Thus, the solution to the differential equation is

y = x - 2 \cos x + 7 .

Reading Questions Check-Point Questions

1. We can solve $\frac{d y}{d x} = x^{3} - 7$ for $y$ by differentiating both sides with respect to $x$ .

We can solve $\frac{d y}{d x} = x^{3} - 7$ for $y$ by differentiating both sides with respect to $x .$

True
Incorrect, we integrate both sides with respect to $x .$
False
Correct!

2. The equation $\frac{d y}{d x} = \ln (3 x + 1)$ implies that the solution, $y,$ is the antiderivative of $\ln (3 x + 1)$ .

The equation $\frac{d y}{d x} = \ln (3 x + 1)$ implies that the solution, $y,$ is the antiderivative of $\ln (3 x + 1) .$

True
Correct, integrating both sides gives

$y = \int \ln (3 x + 1) d x \leftarrow anti-derivative of \ln (3 x + 1) .$
False
Incorrect.

3. Combining constants is a common practice in differential equations..

Combining constants is a common practice in differential equations.

True
Correct!
False
Incorrect, see the footnote above.

4. What is the process to solve for $y$ in the equation $\frac{d y}{d x} = \tan (2 x) ?$

What is the process to solve for $y$ in the equation $\frac{d y}{d x} = \tan (2 x) ?$

Differentiating both sides with respect to $x .$
Incorrect, we integrate both sides with respect to $x .$
Integrating both sides with respect to $x .$
Correct!
Integrating both sides with respect to $y .$
Incorrect, this is not part of the process.
Multiplying both sides by $d x .$
Incorrect, this is not part of the process.

5. How do we start solving the differential equation $\frac{1}{3} y^{'} + 7 x + x^{2} = 1 ?$

How do we start solving the differential equation $\frac{1}{3} y^{'} + 7 x + x^{2} = 1 ?$

By isolating the derivative, $y^{'} .$
Correct! Isolate $y^{'}$ first, then integrate both sides.
By Integrating both sides with respect to $x .$
Incorrect. While you could do this, we suggest isolating the derivative first.
Differentiate both sides with respect to $x .$
Incorrect, we would like to remove derivatives, not add more.
Convert $y^{'}$ to $d y / d x .$
Incorrect, the notation for the derivative does not matter.

You have attempted 1 of 6 activities on this page.

PrevTop Next

Section 3.1 Antiderivatives

Note 1. Combining Constants is very common in differential equations.

Example 2.

Reading Questions Check-Point Questions

1. We can solve dydx=x3−7 for y by differentiating both sides with respect to x.

2. The equation dydx=ln⁡(3x+1) implies that the solution, ,y, is the antiderivative of ln⁡(3x+1).

3. Combining constants is a common practice in differential equations..

4. What is the process to solve for y in the equation ?dydx=tan⁡(2x)?

5. How do we start solving the differential equation ?13y′+7x+x2=1?

1. We can solve $\frac{d y}{d x} = x^{3} - 7$ for $y$ by differentiating both sides with respect to $x$ .

2. The equation $\frac{d y}{d x} = \ln (3 x + 1)$ implies that the solution, $y,$ is the antiderivative of $\ln (3 x + 1)$ .

4. What is the process to solve for $y$ in the equation $\frac{d y}{d x} = \tan (2 x) ?$

5. How do we start solving the differential equation $\frac{1}{3} y^{'} + 7 x + x^{2} = 1 ?$