Integration and the Fundamental Theorem

A structured guide to antiderivatives, substitution, numerical and definite integration, and both parts of the Fundamental Theorem of Calculus.

The role of integration

Integration measures accumulated change and reverses differentiation in an important sense. The main tasks are to find antiderivatives, evaluate definite integrals, approximate integrals numerically, and interpret integrals as signed accumulation.

A useful first question is whether the integral has limits. An expression such as $\int f(x)\,dx$ is an and produces a family of functions. An expression such as $\int_a^b f(x)\,dx$ is a and produces a number.

Core connection

Differentiation describes an instantaneous rate of change. Integration combines infinitely many small contributions to find accumulated change. The makes these operations inverse processes under appropriate continuity conditions.

Takeaway: Identify whether the problem asks for a function, a number, or an approximation before choosing a method.

Antiderivatives and indefinite integrals

An of $f$ is a function $F$ satisfying $F'(x)=f(x)$ . For example, since $\frac{d}{dx}(x^3)=3x^2$ , the function $x^3$ is an of $3x^2$ .

If one is $F(x)$ , then every function $F(x)+C$ , where $C$ is constant, has the same derivative. Therefore,

\int f(x)\,dx=F(x)+C.

The constant is essential because differentiation removes constants. For example,

\int 2x\,dx=x^2+C.

Basic rules

For $n\neq -1$ , the power rule is

\int x^n\,dx=\frac{x^{n+1}}{n+1}+C.

The exponent $n=-1$ is exceptional:

\int \frac{1}{x}\,dx=\ln|x|+C.

Linearity allows terms and constant factors to be handled separately:

\int [f(x)+g(x)]\,dx=\int f(x)\,dx+\int g(x)\,dx,

\int kf(x)\,dx=k\int f(x)\,dx.

Some useful formulas are

\int e^x\,dx=e^x+C,\qquad \int \cos x\,dx=\sin x+C,

\int \sin x\,dx=-\cos x+C,\qquad \int \sec^2x\,dx=\tan x+C.

Example

\begin{aligned} \int \left(6x^2-4x+\frac{3}{x}\right)dx &=2x^3-2x^2+3\ln|x|+C. \end{aligned}

Differentiate the result to check it:

\frac{d}{dx}\left(2x^3-2x^2+3\ln|x|+C\right)=6x^2-4x+\frac{3}{x}.

Takeaway: For an , include $C$ , apply the basic rules term by term, and verify by differentiation.

Substitution and the reverse chain rule

reverses the chain rule. It is useful when an integrand contains an inner expression together with its derivative, or a constant multiple of that derivative.

Procedure

Choose an inner expression and set it equal to $u$ .
Compute $du$ .
Rewrite the entire integral using $u$ and $du$ .
Integrate with respect to $u$ .
Substitute the original expression back for an .
Add $C$ when there are no limits.

For

\int f(g(x))g'(x)\,dx,

set $u=g(x)$ , so that $du=g'(x)\,dx$ . The integral becomes

\int f(u)\,du.

Example with a matching derivative

Let $u=x^2+5$ in

\int 2x(x^2+5)^4\,dx.

Then $du=2x\,dx$ , giving

\int u^4\,du=\frac{u^5}{5}+C.

Substituting back produces

\int 2x(x^2+5)^4\,dx=\frac{(x^2+5)^5}{5}+C.

Example with a constant factor

For

\int \cos(3x)\,dx,

let $u=3x$ . Since $du=3\,dx$ , we have $dx=\frac{1}{3}\,du$ . Thus,

\int \cos(3x)\,dx=\frac{1}{3}\int \cos u\,du=\frac{1}{3}\sin(3x)+C.

The factor $\frac{1}{3}$ compensates for the derivative of the inner expression.

Definite integrals

For limits, either return to the original variable before using the original limits or change the limits with the substitution:

\int_a^b f(g(x))g'(x)\,dx=\int_{g(a)}^{g(b)}f(u)\,du.

For example,

\begin{aligned} \int_0^1 2x(x^2+1)^3\,dx &=\int_1^2u^3\,du\\ &=\left[\frac{u^4}{4}\right]_1^2 =\frac{15}{4}. \end{aligned}

Once the limits have been changed, do not substitute back to $x$ at the end.

Takeaway: A successful substitution accounts for both the new expression and its differential factor.

Approximating integrals numerically

When an exact is unavailable or a problem asks for an approximation, use . Divide $[a,b]$ into $n$ equal subintervals of width

\Delta x=\frac{b-a}{n},

with partition points

x_i=a+i\Delta x.

Rectangle methods

A left uses the left endpoint of each subinterval:

L_n=\Delta x\sum_{i=0}^{n-1}f(x_i).

A right uses the right endpoint:

R_n=\Delta x\sum_{i=1}^{n}f(x_i).

A midpoint sum uses the midpoint $m_i=\frac{x_{i-1}+x_i}{2}$ :

M_n=\Delta x\sum_{i=1}^{n}f(m_i).

Trapezoidal rule

The trapezoidal rule replaces each curved section with a trapezoid:

T_n=\frac{\Delta x}{2}\left[f(x_0)+2f(x_1)+\cdots+2f(x_{n-1})+f(x_n)\right].

For equally spaced points,

T_n=\frac{L_n+R_n}{2}.

If a function is increasing, the left sum generally underestimates and the right sum generally overestimates. For a concave-up function, the trapezoidal rule generally overestimates; for a concave-down function, it generally underestimates. These are tendencies and require attention to changes in behavior.

Example

For $f(x)=x^2$ on $[0,2]$ with two subintervals, $\Delta x=1$ . Therefore,

L_2=1[f(0)+f(1)]=1,

R_2=1[f(1)+f(2)]=5.

The exact value is $\frac{8}{3}$ , which lies between these estimates. The trapezoidal estimate is

T_2=\frac{1+5}{2}=3.

Increasing $n$ usually improves an approximation because the subintervals become narrower.

Takeaway: Choose the numerical method specified by the problem, and use monotonicity or concavity to anticipate the direction of error.

Definite integrals and signed accumulation

A is written

\int_a^b f(x)\,dx

and is defined, when the limit exists, by Riemann sums:

\int_a^b f(x)\,dx=\lim_{n\to\infty}\sum_{i=1}^{n}f(x_i^*)\Delta x.

Here, $x_i^*$ is a sample point in the $i$ -th subinterval. The result measures net signed accumulation: contributions above the horizontal axis are positive, and contributions below it are negative.

Signed area and total area

If $f(x)\geq 0$ throughout $[a,b]$ , the is the geometric area between the graph and the axis. If the function changes sign, the integral gives net signed area, not total geometric area. Total area is found with

\int_a^b|f(x)|\,dx,

usually by splitting at the zeros of $f$ .

For example,

\int_0^{2\pi}\sin x\,dx=0

because the positive and negative contributions cancel. However,

\int_0^{2\pi}|\sin x|\,dx=4.

Important properties

\int_a^a f(x)\,dx=0, \qquad \int_b^a f(x)\,dx=-\int_a^b f(x)\,dx,

\int_a^b[f(x)+g(x)]\,dx=\int_a^b f(x)\,dx+\int_a^b g(x)\,dx,

\int_a^b kf(x)\,dx=k\int_a^b f(x)\,dx,

and for any point $c$ ,

\int_a^b f(x)\,dx=\int_a^c f(x)\,dx+\int_c^b f(x)\,dx.

The variable of integration is a dummy variable, so $\int_a^b f(x)\,dx=\int_a^b f(t)\,dt$ .

Takeaway: Always determine whether an integral represents net signed accumulation or total geometric area.

The two parts of the Fundamental Theorem

The has two complementary parts. Together, they connect accumulation with rates of change.

Accumulation and differentiation

If $f$ is continuous and

F(x)=\int_a^x f(t)\,dt,

then

F'(x)=f(x).

The variable $t$ is a dummy variable inside the integral, while $x$ indicates the changing upper limit. For example,

F(x)=\int_2^x\sqrt{1+t^4}\,dt

has derivative

F'(x)=\sqrt{1+x^4}.

No elementary is needed for this differentiation.

If the upper limit is a function, apply the chain rule:

\frac{d}{dx}\int_a^{g(x)}f(t)\,dt=f(g(x))g'(x).

Thus,

\frac{d}{dx}\int_1^{x^2}\cos(t^3)\,dt=2x\cos(x^6).

If the variable is the lower limit, the sign changes:

\frac{d}{dx}\int_x^a f(t)\,dt=-f(x).

Evaluating a

If $F'(x)=f(x)$ on $[a,b]$ , then

\int_a^b f(x)\,dx=F(b)-F(a).

The notation $[F(x)]_a^b$ means $F(b)-F(a)$ .

For example,

\begin{aligned} \int_1^3(2x+1)\,dx &=[x^2+x]_1^3\\ &=(9+3)-(1+1)=10. \end{aligned}

Net change and motion

If $r(t)$ is the rate of change of a quantity $Q(t)$ , then

Q(b)-Q(a)=\int_a^b r(t)\,dt.

For velocity $v(t)$ , is

s(b)-s(a)=\int_a^b v(t)\,dt,

while total distance is

\int_a^b|v(t)|\,dt.

For $v(t)=t-2$ on $[0,4]$ , is zero because the positive and negative contributions cancel, but total distance is $4$ .

Takeaway: Use Part 1 to differentiate an integral-defined function and Part 2 to evaluate a from an .

A reliable problem-solving strategy

Use this sequence to organize an integration problem:

Classify the problem. Decide whether it is an , a , a numerical approximation, or a derivative of an accumulation function.
Inspect the integrand. Look for a basic , a sum of terms, a power, or a composite expression with a matching derivative.
Select a method. Use direct integration, substitution, a , a numerical rule, or the .
Track constants and limits. Include $C$ only for indefinite integrals. Reversing limits changes the sign.
Interpret signs. Decide whether the result is net signed accumulation, total area, , or total distance.
Check the result. Differentiate an indefinite-integral answer, compare a numerical estimate with expected bounds, or consider the magnitude and sign of a .

Frequent errors

Forgetting $C$ in an .
Applying the power rule to $x^{-1}$ ; instead, $\int x^{-1}\,dx=\ln|x|+C$ .
Omitting the derivative factor in substitution, as in incorrectly writing $\int\cos(3x)\,dx=\sin(3x)+C$ .
Confusing signed area with total geometric area.
Confusing , which uses velocity, with total distance, which uses $|v(t)|$ .
Changing the limits during substitution but then substituting back to the original variable.
Adding $C$ to a , which already produces a number.

A final derivative check is often the fastest way to catch an algebraic or constant-factor error.

The role of integration

Core connection

Antiderivatives and indefinite integrals

Basic rules

Example

Substitution and the reverse chain rule

Procedure

Example with a matching derivative

Example with a constant factor

Definite integrals

Approximating integrals numerically

Rectangle methods

Trapezoidal rule

Example

Definite integrals and signed accumulation

Signed area and total area

Important properties

The two parts of the Fundamental Theorem

Accumulation and differentiation

Evaluating a definite integral

Net change and motion

A reliable problem-solving strategy

Frequent errors

Evaluating a