SME
Calculus I · Problem type
A log result: m ÷ (mx + c)
Use this for “Evaluate the integral of 3 ÷ (3x − 1) from 1 to 2.”
Chart
Shaded: ∫ from 1 to 3 of f(x) dx = 0.8473
All of it is above the axis, so the integral is the shaded area
This is an example. Type your own values to start a new problem.
∫
m
x
+
dx
=
I
u(a)
New lower limit, ma + c
worked out (not typed)
u(b)
New upper limit, mb + c
worked out (not typed)
Formulas
u(
a
)
=
m
×
a
+
c
3 = 2 × 1 + 1
u(
b
)
=
m
×
b
+
c
7 = 2 × 3 + 1
I
=
ln|u(
b
)|
−
ln|u(
a
)|
0.8473 = ln|7| − ln|3|
Assumptions
•
Pick u as the bottom, u = mx + c. Then du = m dx is exactly the top, so the integral is ∫ 1 ÷ u du.
•
The power rule fails for u⁻¹ (it would divide by 0). Instead an antiderivative of 1 ÷ u is ln|u|, so the indefinite answer is ln|mx + c| + C.
•
Change the limits with u: x = a becomes u(a) = ma + c and x = b becomes u(b) = mb + c.
•
The bottom cannot be 0 from a to b: m ≠ 0, and u(a) and u(b) have the same sign. The chart writes m ÷ (mx + c) as 1 ÷ (x + c ÷ m).
Step-by-step
WE KNOW
m = 2, c = 1, a = 1, b = 3
FIND
new lower limit, ma + c (u(a)), new upper limit, mb + c (u(b)), value of the integral (I)
Step 1 · Find new lower limit, ma + c (u(a))
Use u(a) = m × a + c
Put
the
lower
limit
x
=
a
into
the
inside,
u
=
mx
+
c
:
the
integral
in
u
starts
there.
u(
a
)
=
2
×
1
+
1
u(
a
)
=
3
Step 2 · Find new upper limit, mb + c (u(b))
Use u(b) = m × b + c
Put
the
upper
limit
x
=
b
into
the
inside:
the
integral
in
u
ends
there.
u(
b
)
=
2
×
3
+
1
u(
b
)
=
6
+
1
u(
b
)
=
7
Step 3 · Find value of the integral (I)
Use I = ln|u(b)| − ln|u(a)|
Since
du
=
m
dx,
the
top
m
dx
is
du:
the
integral
becomes
the
integral
of
1
u
from
u(
a
)
to
u(
b
).
An
antiderivative
of
1
u
is
ln|u|.
I
=
ln|7|
−
ln|3|
u = 2x + 1, du = 2 dx
3
∫
1
(
2
2
x
+ 1
) dx
=
7
∫
3
(
1
u
) du
7
∫
3
(
1
u
) du
=
[ln|u|]
from
3
to
7
=
0.8473
[ln|2x + 1|] from 1 to 3 = 0.8473
I
=
0.8473
Check
Put the numbers back in. Both sides should match.
3 = 2 × 1 + 1 ✓
7 = 2 × 3 + 1 ✓
0.8473 = ln|7| − ln|3| ✓
Topic check
See what you know: 5 questions on this topic
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