Answered: "Assume h(x) and k(x) have continuous derivatives on..."

High School
Other

Joanna Benson

Answered question

2021-12-11

Assume h(x) and k(x) have continuous derivatives on [a, b] and derive the integration-by-parts formula
$\int_{a}^{b} h (t) k^{'} (t) d t = h (b) k (b) - h (a) k (a) - \int_{a}^{b} h^{'} (t) k (t) d t$

Answer & Explanation

amarantha41

Beginner2021-12-12Added 38 answers

Step 1
Let u,v be two functions of t
Using product rule
d(uv)=udv+vdu
Integrating on both sides
$\int d (u v) = \int u d v + v d u$
$u v = \int u d v + \int v d u$
$\int u d v = u v - \int v d u$
Step 2
Consider
$\int_{a}^{b} h (t) k^{'} (t) d t$
Let
u=h(t)
dv=k'(t)=> v=k(t)
Using integration by parts
$\int_{a}^{b} h (t) k^{'} (t) d t = [h (t) k (t)]_{a}^{b} - \int_{a}^{b} k (t) h^{'} (t) d t$
$\int_{a}^{b} h (t) k^{'} (t) d t = h (b) k (b) - h (a) k (a) - \int_{a}^{b} k (t) h^{'} (t) d t$

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Other

22+64
When a cold drink is taken from a refrigerator, its temperature is 5 degree C. After 25 minutes in a 20 degree C room its temperature has increased to 10 degree C. What is the temperature of the drink after 50 minutes?
How many minutes are there in $2 \frac{3}{5}$ hours?
Find the answer for $64$ power $\frac{1}{6}$ .
The following law describes the relationship between gas volume and pressure: ________. A)Boyle's law; B)Henry's law; C)Charles' law; D)Dalton's law
If an electric switch is open then no current flows through the circuit.
A)True;
B)False
The price elasticity of supply of a good is 0.8. Its price rises by 50 per cent. Calculate the percentage increase in its supply.
How many quarts are in 10 gallons?
When the net force acting on an object is zero, the total momentum of the system will:
A)remain constant;
B)increase with time;
C)decrease with time;
D)None of these
How many centimeters are in four meters?
A notebook computer has a mass of 2.25 kilograms. About how many pounds does the notebook weigh?
If the dot product of two non-zero vectors is zero, then the vectors A)are parallel to each other. B)are perpendicular to each other. C)can have any orientation. D)are anti-parallel to each other.
Which is longer 5 miles or 10 kilometers?
9 grams is how many milligrams?
How to write $673.5$ in scientific notation?

Didn't find what you were looking for?