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Mathematics
Mathematics
Problem
Evaluate the following integral, showing all necessary steps: a. (i) Show that
∫
2
x
cos
(
x
2
)
d
x
=
sin
(
x
2
)
+
C
.
\displaystyle \int 2x\cos\bigl(x^{2}\bigr)\,dx = \sin\bigl(x^{2}\bigr)+C.
∫
2
x
cos
(
x
2
)
d
x
=
sin
(
x
2
)
+
C
.
b. (ii) Evaluate
∫
1
x
sin
(
ln
x
)
d
x
.
\displaystyle \int \frac{1}{x}\,\sin(\ln x)\,dx.
∫
x
1
sin
(
ln
x
)
d
x
.
c. (iii) Hence, determine
∫
(
2
x
cos
(
x
2
)
−
1
x
sin
(
ln
x
)
)
d
x
.
\displaystyle \int \Bigl(2x\cos\bigl(x^{2}\bigr)-\tfrac{1}{x}\,\sin(\ln x)\Bigr)\,dx.
∫
(
2
x
cos
(
x
2
)
−
x
1
sin
(
ln
x
)
)
d
x
.
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