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Given a real valued function which is monotonic and differentiable
MATHEMATICS · Subjective · ISC · Mathematics · Physics · Calculus
Problem
Given a real valued function
f
(
x
)
f(x)
f
(
x
)
which is monotonic and differentiable, prove that for any real numbers
a
a
a
and
b
b
b
;
∫
a
b
{
f
2
(
x
)
−
f
2
(
a
)
}
d
x
=
∫
f
(
a
)
f
(
b
)
2
x
{
b
−
f
−
1
(
x
)
}
d
x
\int_{a}^{b} \{f^2(x) - f^2(a)\} dx = \int_{f(a)}^{f(b)} 2x \{b - f^{-1}(x)\} dx
∫
a
b
{
f
2
(
x
)
−
f
2
(
a
)}
d
x
=
∫
f
(
a
)
f
(
b
)
2
x
{
b
−
f
−
1
(
x
)}
d
x
Related theory
Integration by Parts
Integration Techniques
Properties of Definite Integrals
Integration by Substitution
Exponential and Logarithmic Functions
Differentiation Rules
Second fundamental theorem of calculus
Invertible functions and inverse of a function
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