SME
Calculus I · Problem type
A linear inside: (mx + c)ⁿ
Use this for “Evaluate the integral of (3x − 2)² from 1 to 2.”
Chart
Shaded: ∫ from 0 to 1 of f(x) dx = 10
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.
∫
(
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
1 = 2 × 0 + 1
u(
b
)
=
m
×
b
+
c
3 = 2 × 1 + 1
I
=
(u
(
b
)
n
+ 1
−
u
(
a
)
n
+ 1
)
÷
(
m
×
(
n
+
1))
10
=
3
3 + 1
−
1
3 + 1
2 × (3 + 1)
Assumptions
•
Pick u as the linear inside, u = mx + c. Then du = m dx, so dx = du ÷ m: the outside needs no factor of x.
•
Change the limits with u: x = a becomes u(a) = ma + c and x = b becomes u(b) = mb + c.
•
In u the integral is (1 ÷ m) × ∫ uⁿ du, so an antiderivative is uⁿ⁺¹ ÷ (m(n + 1)). Putting u back gives (mx + c)ⁿ⁺¹ ÷ (m(n + 1)), the indefinite answer up to + C.
•
The slope m cannot be 0, and area below the x-axis counts as negative.
Step-by-step
WE KNOW
m = 2, c = 1, n = 3, a = 0, b = 1
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
×
0
+
1
u(
a
)
=
1
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
×
1
+
1
u(
b
)
=
3
Step 3 · Find value of the integral (I)
Use I = (u(b)ⁿ⁺¹ − u(a)ⁿ⁺¹) ÷ (m × (n + 1))
Since
du
=
m
dx,
dx
is
du
m
:
the
integral
becomes
the
integral
of
uⁿ
from
u(
a
)
to
u(
b
),
divided
by
m
.
Raise
the
power
by
1,
then
divide
by
the
new
power
and
by
m
.
I
=
3
4
−
1
4
8
u = 2x + 1, du = 2 dx
1
∫
0
(2
x
+ 1)
3
dx
=
3
∫
1
(
u
3
2
) du
3
∫
1
(
u
3
2
) du
=
[
u
4
8
]
from
1
to
3
=
10
I
=
10
Check
Put the numbers back in. Both sides should match.
1 = 2 × 0 + 1 ✓
3 = 2 × 1 + 1 ✓
10
=
[
(2x + 1)
4
÷
8]
from
0
to
1
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.