Learn how to evaluating improper integrals problems with Infinite Discontinuity, Practice Type 1 improper integral problems, infinite discontinuity calculus functions.
While evaluating \(\displaystyle \int\limits^b_a f(x)dx\) there was an assumption that
(1) f does not have an infinite discontinuity in \([a, b]\) .
(2) The integral \([a, b]\) is not infinite i.e, both \(a\) and \(b\) are finite values.
\(\displaystyle \int\limits^\infty_a f(x)dx= \lim_{t \to \infty} \displaystyle \int\limits^t_a f(x)dx\)
is a finite value
and \(\displaystyle \int\limits^b_{-\infty} f(x)dx= \lim_{t \to -\infty} \displaystyle \int\limits^b_t f(x)dx\)
is a finite value
and divergent if corresponding limit does not exists.
A \(I=\displaystyle \int\limits^4_2 (x+3)dx\)
B \(I=\displaystyle \int\limits^5_{-2} (x^2+5)dx\)
C \(I=\displaystyle \int\limits^7_{-5} (x^2+5)dx\)
D \(I=\displaystyle \int\limits^\infty_2 \dfrac{1}{x^2}dx\)
While evaluating \(\displaystyle \int\limits^b_a f(x)dx\) there was an assumption that
(1) f does not have an infinite discontinuity in \([a, b]\) .
(2) The integral \([a, b]\) is not infinite i.e, both \(a\) and \(b\) are finite values.
These improper integral \(\displaystyle \int\limits^\infty_a f(x)dx\) and \(\displaystyle \int\limits^b_{-\infty} f(x)dx\) are called convergent if corresponding limit exists.
By existence of limit we mean that
\(\displaystyle \int\limits^\infty_a f(x)dx= \lim_{t \to \infty} \displaystyle \int\limits^t_a f(x)dx\)
is a finite value
and \(\displaystyle \int\limits^b_{-\infty} f(x)dx= \lim_{t \to -\infty} \displaystyle \int\limits^b_t f(x)dx\)
is a finite value
and divergent if corresponding limit does not exists.
A \(I=\displaystyle \int\limits^{-1}_{-2} \dfrac{1}{x^5}dx\)
B \(I=\displaystyle \int\limits^{-2}_{-4} \dfrac{1}{x^2}dx\)
C \(I=\displaystyle \int\limits^2_{–\,\infty} \dfrac{1}{x^2}dx\)
D \(I=\displaystyle \int\limits^{-1}_{-5} \dfrac{1}{x^3}dx\)
While evaluating \(\displaystyle \int\limits^b_a f(x)dx\) there was an assumption that
(1) f does not have an infinite discontinuity in \([a, b]\) .
(2) The integral \([a, b]\) is not infinite i.e, both \(a\) and \(b\) are finite values.
\(\displaystyle \int\limits^\infty_a f(x)dx= \lim_{t \to \infty} \displaystyle \int\limits^t_a f(x)dx\)
is a finite value
and \(\displaystyle \int\limits^b_{-\infty} f(x)dx= \lim_{t \to -\infty} \displaystyle \int\limits^b_t f(x)dx\)
is a finite value
and divergent if corresponding limit does not exists.
A \(-\dfrac{1}{9}\)
B \(\dfrac{1}{2}\)
C \(\dfrac{1}{3}\)
D \(-\dfrac{1}{6}\)
While evaluating \(\displaystyle \int\limits^b_a f(x)dx\) there was an assumption that
(1) f does not have an infinite discontinuity in \([a, b]\) .
(2) The integral \([a, b]\) is not infinite i.e, both \(a\) and \(b\) are finite values.
\(\displaystyle \int\limits^\infty_a f(x)dx= \lim_{t \to \infty} \displaystyle \int\limits^t_a f(x)dx\)
is a finite value
and \(\displaystyle \int\limits^b_{-\infty} f(x)dx= \lim_{t \to -\infty} \displaystyle \int\limits^b_t f(x)dx\)
is a finite value
and divergent if corresponding limit does not exists.
If both \(\displaystyle \int\limits^a_{-\infty} f(x)dx\) and \(\displaystyle \int\limits^\infty_a f(x)dx\) are convergent then
\(\displaystyle \int\limits^\infty_{-\infty} f(x)dx=\displaystyle \int\limits^a_{-\infty} f(x)dx+\displaystyle \int\limits^\infty_a f(x)dx\)
where 'a' can be any real number.
A \(\dfrac{\pi}{10}\)
B \(\dfrac{\pi}{5}\)
C \(\dfrac{\pi^2}{3}\)
D \(\pi\)
While evaluating \(\displaystyle \int\limits^b_a f(x)dx\) there was an assumption that
(1) f does not have an infinite discontinuity in \([a, b]\) .
(2) The integral \([a, b]\) is not infinite i.e, both \(a\) and \(b\) are finite values.
These improper integral \(\displaystyle \int\limits^\infty_a f(x)dx\) and \(\displaystyle \int\limits^b_{-\infty} f(x)dx\) are called convergent if corresponding limit exists.
By existence of limit we mean that
\(\displaystyle \int\limits^\infty_a f(x)dx= \lim_{t \to \infty} \displaystyle \int\limits^t_a f(x)dx\)
is a finite value
and \(\displaystyle \int\limits^b_{-\infty} f(x)dx= \lim_{t \to -\infty} \displaystyle \int\limits^b_t f(x)dx\)
is a finite value
and divergent if corresponding limit does not exists.
A \(\dfrac{1}{2}\)
B \(\dfrac{1}{3}\)
C \(-\dfrac{1}{2}\)
D \(\dfrac{1}{4}\)
While evaluating \(\displaystyle \int\limits^b_a f(x)dx\) there was an assumption that
(1) f does not have an infinite discontinuity in \([a, b]\) .
(2) The integral \([a, b]\) is not infinite i.e, both \(a\) and \(b\) are finite values.
\(\displaystyle \int\limits^\infty_a f(x)dx= \lim_{t \to \infty} \displaystyle \int\limits^t_a f(x)dx\)
is a finite value
and \(\displaystyle \int\limits^b_{-\infty} f(x)dx= \lim_{t \to -\infty} \displaystyle \int\limits^b_t f(x)dx\)
is a finite value
and divergent if corresponding limit does not exists.
A \(\dfrac{\pi}{4}\)
B \(\dfrac{1}{2}\)
C \(\dfrac{\pi}{6}\)
D \(\dfrac{\pi}{8}\)
While evaluating \(\displaystyle \int\limits^b_a f(x)dx\) there was an assumption that
(1) f does not have an infinite discontinuity in \([a, b]\) .
(2) The integral \([a, b]\) is not infinite i.e, both \(a\) and \(b\) are finite values.
\(\displaystyle \int\limits^\infty_a f(x)dx= \lim_{t \to \infty} \displaystyle \int\limits^t_a f(x)dx\)
is a finite value
and \(\displaystyle \int\limits^b_{-\infty} f(x)dx= \lim_{t \to -\infty} \displaystyle \int\limits^b_t f(x)dx\)
is a finite value
and divergent if corresponding limit does not exists.
For a positive function \(f\) we know that
\(\displaystyle \int\limits^b_a f(x)dx\) = Area bounded by the curve
\(y=f(x)\), lines \(x=a,\;x=b\) and \(x\) axis.
If \(\displaystyle \int\limits^\infty_a f(x)dx\) is convergent we say that it is the area of region such that \(x \geq a\) and \(0 \leq y \leq f(x)\)
\(A(S)=\displaystyle \int\limits^\infty_a f(x)dx\)
Note that the area bounded by \(y=f(x)\), \(x\) axis,
\(x=a\) and \(x=b\) is given by
\(\displaystyle \int\limits^b_a f(x)dx\)
In this case \(b \to \infty\).
A \(A(S)=2\)
B \(A(S)=10\)
C \(A(S)=\dfrac{1}{2}\)
D \(A(S)=1\)
For a positive function \(f\) we know that,
\(\displaystyle \int\limits^b_a f(x)dx\) = Area bounded by the curve
\(y=f(x)\), lines \(x=a,\;x=b\) and \(x\) axis.
\(A(S)=\displaystyle \int\limits^\infty_a f(x)dx\)
\(x=a\) and \(x=b\) is given by
\(\displaystyle \int\limits^b_a f(x)dx\)
In this case \(b \to \infty\)
A \(A(S)=\dfrac{1}{4}\)
B \(A(S)=\dfrac{1}{2}\)
C \(A(S)=\dfrac{1}{10}\)
D \(A(S)=5\)