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Joint Entrance Examination
Mathematics
Indefinite Integrals
Question 318
Question 318 of 416
Medium
MCQ
If
∫
d
x
x
+
x
7
=
p
(
x
)
\displaystyle\int \dfrac{dx}{x+x^7} = p(x)
∫
x
+
x
7
d
x
=
p
(
x
)
then
∫
x
6
x
+
x
7
d
x
\displaystyle\int \dfrac{x^6}{x+x^7}\,dx
∫
x
+
x
7
x
6
d
x
is equal to:
A
ln
∣
x
∣
−
p
(
x
)
+
c
\ln|x|-p(x)+c
ln
∣
x
∣
−
p
(
x
)
+
c
B
ln
∣
x
∣
+
p
(
x
)
+
c
\ln|x|+p(x)+c
ln
∣
x
∣
+
p
(
x
)
+
c
C
x
−
p
(
x
)
+
c
x-p(x)+c
x
−
p
(
x
)
+
c
D
x
+
p
(
x
)
+
c
x+p(x)+c
x
+
p
(
x
)
+
c
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Official solution
∫
x
6
x
+
x
7
d
x
=
∫
x
6
x
(
1
+
x
6
)
d
x
=
∫
(
1
+
x
6
)
−
1
x
(
1
+
x
6
)
d
x
\int \dfrac{x^6}{x+x^7}\,dx = \int \dfrac{x^6}{x(1+x^6)}\,dx = \int \dfrac{(1+x^6)-1}{x(1+x^6)}\,dx
∫
x
+
x
7
x
6
d
x
=
∫
x
(
1
+
x
6
)
x
6
d
x
=
∫
x
(
1
+
x
6
)
(
1
+
x
6
)
−
1
d
x
=
∫
1
x
d
x
−
∫
1
x
+
x
7
d
x
=
ln
∣
x
∣
−
p
(
x
)
+
c
= \int \dfrac1x\,dx - \int \dfrac{1}{x+x^7}\,dx = \ln|x|-p(x)+c
=
∫
x
1
d
x
−
∫
x
+
x
7
1
d
x
=
ln
∣
x
∣
−
p
(
x
)
+
c
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