Calculus

Integration of Functions

Improper Integrals

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Problem 1
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Problems 2-10

The definite integral \(\int\limits_a^b {f\left( x \right)dx} \) is called an improper integral if one of two situations occurs:

  • The limit \(a\) or \(b\) (or both the limits) are infinite;
  • The function \({f\left( x \right)}\) has one or more points of discontinuity in the interval \(\left[ {a,b} \right].\)

Infinite Limits of Integration

Let \({f\left( x \right)}\) be a continuous function on the interval \(\left[ {a,\infty} \right).\) We define the improper integral as

\[{\int\limits_a^\infty {f\left( x \right)dx} }={ \lim\limits_{n \to \infty } \int\limits_a^n {f\left( x \right)dx} .}\]

Consider the case when \({f\left( x \right)}\) is a continuous function on the interval \(\left( {-\infty, b} \right].\) Then we define the improper integral as

\[{\int\limits_{ – \infty }^b {f\left( x \right)dx} }={ \lim\limits_{n \to – \infty } \int\limits_n^b {f\left( x \right)dx} .}\]

If these limits exist and are finite then we say that the improper integrals are convergent. Otherwise the integrals are divergent.

Let \({f\left( x \right)}\) be a continuous function for all real numbers. We define:

\[
{\int\limits_{ – \infty }^\infty {f\left( x \right)dx} }
= {\int\limits_{ – \infty }^c {f\left( x \right)dx} }+{ \int\limits_c^\infty {f\left( x \right)dx} .}
\]

If, for some real number \(c,\) both of the integrals in the right side are convergent, then we say that the integral \(\int\limits_{ – \infty }^\infty {f\left( x \right)dx} \) is also convergent; otherwise it is divergent.

Comparison Theorems

Let \({f\left( x \right)}\) and \({g\left( x \right)}\) be continuous functions on the interval \(\left[ {a,\infty } \right).\) Suppose that \(0 \le g\left( x \right) \le f\left( x \right)\) for all \(x\) in the interval \(\left[ {a,\infty } \right).\)

  1. If \(\int\limits_a^\infty {f\left( x \right)dx} \) is convergent, then \(\int\limits_a^\infty {g\left( x \right)dx} \) is also convergent;
  2. If \(\int\limits_a^\infty {g\left( x \right)dx} \) is divergent, then \(\int\limits_a^\infty {f\left( x \right)dx} \) is also divergent;
  3. If \(\int\limits_a^\infty {\left| {f\left( x \right)} \right|dx} \) is convergent, then \(\int\limits_a^\infty {f\left( x \right)dx} \) is also convergent. In this case, we say that the integral \(\int\limits_a^\infty {f\left( x \right)dx} \) is absolutely convergent.

Discontinuous Integrand

Let \({f\left( x \right)}\) be a function which is continuous on the interval \(\left[ {a,b} \right),\) but is discontinuous at \(x = b.\) We define the improper integral as

\[{\int\limits_a^b {f\left( x \right)dx} }={ \lim\limits_{\tau \to 0 + } \int\limits_a^{b – \tau } {f\left( x \right)dx} .}\]

Similarly we can consider the case when the function \({f\left( x \right)}\) is continuous on the interval \(\left( {a,b} \right],\) but is discontinuous at \(x = a.\) Then

\[{\int\limits_a^b {f\left( x \right)dx} }={ \lim\limits_{\tau \to 0 + } \int\limits_{a + \tau }^b {f\left( x \right)dx} .}\]

If these limits exist and are finite then we say that the integrals are convergent; otherwise the integrals are divergent.

Let \({f\left( x \right)}\) be a continuous function for all real numbers \(x\) in the interval \(\left[ {a,b} \right],\) except for some point \(c \in \left( {a,b} \right).\) We define:

\[
{\int\limits_a^b {f\left( x \right)dx} }
= {\int\limits_a^c {f\left( x \right)dx} }+{ \int\limits_c^b {f\left( x \right)dx} ,}
\]

and say that the integral \(\int\limits_a^b {f\left( x \right)dx}\) is convergent if both of the integrals in the right side are also convergent. Otherwise the improper integral is divergent.

Solved Problems

Click on problem description to see solution.

 Example 1

Determine for what values of \(k\) the integral \({\int\limits_1^\infty} {\large\frac{{dx}}{{{x^k}}}\normalsize}\;\left( {k > 0,k \ne 1} \right)\) converges.

 Example 2

Calculate the integral \(\int\limits_0^\infty {\large\frac{{dx}}{{{x^2} + 16}}\normalsize} .\)

 Example 3

Determine whether the integral \({\int\limits_1^\infty} {\large\frac{{dx}}{{{x^2}{e^x}}}\normalsize}\) converges or diverges?

 Example 4

Calculate the integral \({\int\limits_{ – 2}^2} {\large\frac{{dx}}{{{x^3}}}\normalsize}.\)

 Example 5

Determine whether the improper integral \({\int\limits_{ – \infty }^\infty} {\large\frac{{dx}}{{{x^2} + 2x + 8}}\normalsize}\) converges or diverges?

 Example 6

Determine whether the integral \({\int\limits_1^\infty} {{\large\frac{{\sin x}}{{\sqrt {{x^3}} }}\normalsize} dx}\) converges or diverges?

 Example 7

Determine whether the integral \({\int\limits_0^4} {\large\frac{{dx}}{{{{\left( {x – 2} \right)}^3}}}\normalsize}\) converges or diverges?

 Example 8

Determine for what values of \(k\) the integral \({\int\limits_0^1} {\large\frac{{dx}}{{{x^k}}}\normalsize}\) \(\left( {k \gt 0,k \ne 1} \right)\) converges?

 Example 9

Find the area above the curve \(y = \ln x\) in the lower half-plane between \(x = 0\) and \(x = 1.\)

 Example 10

Find the circumference of the unit circle.

Example 1.

Determine for what values of \(k\) the integral \({\int\limits_1^\infty} {\large\frac{{dx}}{{{x^k}}}\normalsize}\;\left( {k > 0,k \ne 1} \right)\) converges.

Solution.

By the definition of an improper integral, we have

\[
{\int\limits_1^\infty {\frac{{dx}}{{{x^k}}}} }
= {\lim\limits_{n \to \infty } \int\limits_1^n {\frac{{dx}}{{{x^k}}}} }
= {\lim\limits_{n \to \infty } \int\limits_1^n {{x^{ – k}}dx} }
= {\lim\limits_{n \to \infty } \left. {\left( {\frac{{{x^{ – k + 1}}}}{{ – k + 1}}} \right)} \right|_1^n }
= {\frac{1}{{1 – k}} \cdot \lim\limits_{n \to \infty } \left. {\left( {{x^{ – k + 1}}} \right)} \right|_1^n }
= {\frac{1}{{1 – k}} \cdot \lim\limits_{n \to \infty } \left( {{n^{ – k + 1}} – {1^{ – k + 1}}} \right) }
= {\frac{1}{{k – 1}} \cdot \lim\limits_{n \to \infty } \left( {1 – {n^{1 – k}}} \right).}
\]

As seen from the expression, there are \(2\) cases:

  • If \(0 \lt k \lt 1,\) then \({n^{1 – k}} \to \infty \) as \(n \to \infty\) and the integral diverges;
  • If \(k \gt 1,\) then \({n^{1 – k}}\) \( = {\large\frac{1}{{{n^{k – 1}}}\normalsize}} \to 0\) as \(n \to \infty\) and the integral converges.
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Problem 1
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Problems 2-10