In the following exercises, solve for the antiderivative ∫f of f with C=0, then use a calculator

In the following exercises, solve for the antiderivative ∫f...

In the following exercises, solve for the antiderivative $\int f$ of $f$ with $C=0,$ then use a calculator to graph $f$ and the antiderivative over the given interval $[a, b] .$ Identify a value of $C$ such that adding $C$ to the antiderivative recovers the definite integral $F(x)=\int_{a}^{x} f(t) d t .$ $$[\mathbf{T}] \int \frac{9}{9+x^{2}} d x \text { over } [-6,6]$$

In the following exercises, solve for the antiderivative $\int f$ of $f$ with $C=0,$ then use a calculator to graph $f$ and the antiderivative over the given interval $[a, b] .$ Identify a value of $C$ such that adding $C$ to the antiderivative recovers the definite integral $F(x)=\int_{a}^{x} f(t) d t .$
$$[\mathbf{T}] \int \frac{9}{9+x^{2}} d x \text { over } [-6,6]$$

Key Concepts

Inverse Trigonometric Integration

Inverse trigonometric integration involves the use of standard formulas to integrate functions that are of the form 1/(a^2+x^2). A common result is that the integral of 1/(a^2+x^2) is (1/a)arctan(x/a), up to an added constant, which is particularly relevant when dealing with rational functions that lead to inverse trigonometric functions upon integration.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration by stating that integration can be reversed by differentiation and vice versa. It allows the evaluation of a definite integral by finding an antiderivative of the function and computing its difference at the endpoints, thereby linking the indefinite integral with the definite integral.

Antiderivatives

An antiderivative of a function is another function whose derivative is the original function. In the context of calculus, finding the antiderivative is synonymous with computing the indefinite integral of the function, and it represents the most general form of the original function without any initial conditions applied.

Constant of Integration

When computing an indefinite integral, a constant of integration is added to account for all possible antiderivatives that differ by a constant. This constant is crucial when working with initial value problems or when aligning an antiderivative with a definite integral that has specific boundary conditions.

Transcript

00:01 So i asked to find the definite integral of the following function, which is 9 over 9 plus x squared dx.

00:09 And this is from negative 6 to 6.

00:11 So the first thing we're going to do is find the corresponding indefinite integral.

00:16 So we'll go ahead and do that together.

00:19 So let's go ahead and do this on the next page.

00:21 We'll do this in blue.

00:24 So i'll go ahead and rewrite this.

00:26 But the first thing we're going to do is apply the multiple constant rule.

00:30 So this turns into 9 times the integral.

00:33 Of 1 over x squared plus 9 dx great now we're going to go ahead and do some substitutions so the first substitution i'm going to do is for x i'm going to say let x equal to 3 u thus i can say that d x is equal to 3 times d u and that follows that u is equal to x over 3 so this integral becomes 9 times the integral of 1 over 3 times u squared plus 1 d u so now we're going to go ahead and apply the constant multiple rule uh once again by taking out this 1 3rd so this turns into 9 times 1 3rd times the integral of 1 over u squared plus 1 u u u.

01:32 Great so we know that the integral of 1 over u square plus 1 is familiar to us it is arc tangent so this actually turns into three times the inverse tangent or arc tangent of you so now what we have to do is recall i'll do this part in black we're going to recall what our value for u is x over 3 so this turns into three times the inverse tangent of x over three.

02:10 The last thing you have to do is apply our constant of integration, and it looks something like this...

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