Using Properties of Definite Integrals Given ∫ − 1 1 f ( x ) d x = 0 and ∫ 0 1 f ( x ) d x = 5 , evaluate (a) ∫ − 1 0 f ( x ) d x . (b) ∫ 0 1 f ( x ) d x − ∫ − 1 0 f ( x ) d x . (c) ∫ − 1 1 3 f ( x ) d x . (d) ∫ 0 1 3 f ( x ) d x .
Using Properties of Definite Integrals Given ∫ − 1 1 f ( x ) d x = 0 and ∫ 0 1 f ( x ) d x = 5 , evaluate (a) ∫ − 1 0 f ( x ) d x . (b) ∫ 0 1 f ( x ) d x − ∫ − 1 0 f ( x ) d x . (c) ∫ − 1 1 3 f ( x ) d x . (d) ∫ 0 1 3 f ( x ) d x .
Solution Summary: The author explains how to calculate a definite integral using the provided values. The additive interval property is: if f(x) is integrable on the three closed intervals determined by
Using Properties of Definite Integrals Given
∫
−
1
1
f
(
x
)
d
x
=
0
and
∫
0
1
f
(
x
)
d
x
=
5
, evaluate
(a)
∫
−
1
0
f
(
x
)
d
x
.
(b)
∫
0
1
f
(
x
)
d
x
−
∫
−
1
0
f
(
x
)
d
x
.
(c)
∫
−
1
1
3
f
(
x
)
d
x
.
(d)
∫
0
1
3
f
(
x
)
d
x
.
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
Reduction formulas Use the reduction formulas in a table of integrals to evaluate the following integrals.
∫x3 e2x dx
Applying reduction formulas Use the reduction formulas in evaluate the following integrals.
∫x3sin x dx
a. Explain why
0 ≤ x²arctan(x) ≤ (pi*x²)/4 for all 0 ≤ x ≤ 1.
b. Use the properties of the integrals to show that the value of the integral
lower bound is 0, higher bound is 1 and the integral is x² arctan(x) dx
lies on the interval [0,pi/12]
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Numerical Integration Introduction l Trapezoidal Rule Simpson's 1/3 Rule l Simpson's 3/8 l GATE 2021; Author: GATE Lectures by Dishank;https://www.youtube.com/watch?v=zadUB3NwFtQ;License: Standard YouTube License, CC-BY