Evaluate
ate the
1-sin x
sin x COS X
TT
dx 26. s
4
25.
1 - cos x
0
cos* x + sin* x
2
Answers
Answered by
1
Answer:
J=∫
1+sinx+cosx
sin
2
x+sinx
Put u=tan(
2
x
)⇒du=
2
1
sec
2
xdx
J=∫
(u
2
+1)(
u
2
+1
2u
+
u
2
+1
1−u
2
+1)
2(
(u
2
+1)
2
4u
2
+
u
2
+1
2u
)
du
=2∫(
(u
2
+1)
2
u−1
+
u
2
+1
1
)du=2tan
−1
u+2∫
(u
2
+1)
2
u−1
du
Put u=tant⇒du=⇒sec
2
tdt
∴J=2tan
−1
u+2∫cos
2
t(tant−1)dt
=2tan
−1
u+2∫(1−sin
2
t)(tant−1)
=2tan
−1
u−cos(tan
−1
u)2−sin(tan
−1
u)cos(tan
−1
u)
=
2
1
(x−sinx−cosx)
K=∫
1+cosx+sinx
cos
2
x+cosx
dx
Put u=tan
2
x
⇒du=
2
1
sec
2
2
x
dx
K=−∫
u
2
+2u
2
+1
2(u−1)
du
Put u=tant⇒du=sec
2
tdt
∴k=−∫cos
2
t(tant−1)dt
=tan
−1
u+cos(tan
−1
u)
2
+sin(tan
−1
u)cos(tan
−1
u)
∴J=K−(sinx+cosx)+c
Step-by-step explanation:
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