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Byju's Answer
Standard XII
Mathematics
Property 4
∫ a b f x f x...
Question
∫
a
b
f
x
f
x
+
f
a
+
b
-
x
d
x
.
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Solution
Let
I
=
∫
a
b
f
x
f
x
+
f
a
+
b
-
x
d
x
.
.
.
(
i
)
=
∫
a
b
f
a
+
b
-
x
f
a
+
b
-
x
+
f
a
+
b
-
a
-
b
+
x
d
x
=
∫
a
b
f
a
+
b
-
x
f
a
+
b
-
x
+
f
x
d
x
∴
I
=
∫
a
b
f
a
+
b
-
x
f
x
+
f
a
+
b
-
x
d
x
.
.
.
(
ii
)
Adding
(
i
)
and
(
ii
)
we
get
2
I
=
∫
a
b
f
x
f
x
+
f
a
+
b
-
x
+
f
a
+
b
-
x
f
x
+
f
a
+
b
-
x
d
x
=
∫
a
b
f
x
+
f
a
+
b
-
x
f
x
+
f
a
+
b
-
x
dx
=
x
a
b
=
b
-
a
Hence
,
I
=
b
-
a
2
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Similar questions
Q.
If
∫
b
a
f
(
x
)
f
(
x
)
+
f
(
a
+
b
−
x
)
dx
=
10
, then which values a and b can take?
