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Math Problems
Calculus
Find indefinite integrals using the substitution and by parts
Find integral:
∫
x
(
x
)
d
x
\int x^{(x)}\,dx
∫
x
(
x
)
d
x
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Solve for
x
x
x
and graph the solution on the number line below.
\newline
x
−
2
<
0
\frac{x}{-2} < 0
−
2
x
<
0
\newline
Answer Attempt
1
1
1
out of
2
2
2
\newline
<
<
<
\newline
≥
\geq
≥
\newline
≥
\geq
≥
or
\newline
Inequality Notation:
\newline
Number Line:
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The volume of the solid obtained by rotating the region enclosed by
\newline
y
=
x
2
,
x
=
y
2
y=x^{2}, \quad x=y^{2}
y
=
x
2
,
x
=
y
2
\newline
about the line
x
=
−
4
x=-4
x
=
−
4
can be computed using the method of disks or washers via an integral
\newline
V
=
∫
a
b
V=\int_{a}^{b}
V
=
∫
a
b
\newline
\newline
with limits of integration
a
=
a=
a
=
and
b
=
b=
b
=
\newline
The volume is
V
=
V=
V
=
cubic units.
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∫
d
x
3
x
−
1
−
4
x
−
1
3
\int \frac{d x}{3 \sqrt{x-1}-4 \sqrt[3]{x-1}}
∫
3
x
−
1
−
4
3
x
−
1
d
x
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∫
cos
x
d
x
sin
2
x
+
4
sin
x
−
1
\int \frac{\cos x d x}{\sqrt{\sin ^{2} x+4 \sin x-1}}
∫
s
i
n
2
x
+
4
s
i
n
x
−
1
c
o
s
x
d
x
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Integration techniques: Evaluate
∫
(
x
2
+
3
x
−
2
)
d
x
\int (x^2 + 3x - 2) \, dx
∫
(
x
2
+
3
x
−
2
)
d
x
using appropriate integration methods.
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∫
x
d
x
+
∫
2
d
x
\int x d x+\int 2 d x
∫
x
d
x
+
∫
2
d
x
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Solve the equation. Check your solution
\newline
19
=
2
−
(
z
+
5
)
19=2-(z+5)
19
=
2
−
(
z
+
5
)
\newline
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∫
x
9
−
4
x
2
d
x
\int \frac{x}{\sqrt{9-4 x^{2}}} d x
∫
9
−
4
x
2
x
d
x
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Calculate the integral and write your answer in simplest form.
\newline
∫
x
5
4
d
x
\int \frac{\sqrt{x^{5}}}{4} \mathrm{dx}
∫
4
x
5
dx
\newline
Answer:
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Calculate the integral and write your answer in simplest form.
\newline
∫
5
x
3
4
d
x
\int \frac{5 \sqrt{x^{3}}}{4} \mathrm{dx}
∫
4
5
x
3
dx
\newline
Answer:
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Calculate the integral and write your answer in simplest form.
\newline
∫
3
x
5
2
d
x
\int \frac{3 \sqrt{x^{5}}}{2} \mathrm{dx}
∫
2
3
x
5
dx
\newline
Answer:
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∫
cos
(
12
r
)
cos
(
15
r
)
d
r
\int \cos(12r) \cos(15r) \, dr
∫
cos
(
12
r
)
cos
(
15
r
)
d
r
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∫
(
x
−
3
)
d
x
x
+
3
\int \frac{(x-3)\,dx}{x+3}
∫
x
+
3
(
x
−
3
)
d
x
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Let
f
f
f
be the function defined by
f
(
x
)
=
4
x
f(x)=4 \sqrt{x}
f
(
x
)
=
4
x
. If three subintervals of equal length are used, what is the value of the right Riemann sum approximation for
∫
0
9
4
x
d
x
\int_{0}^{9} 4 \sqrt{x} d x
∫
0
9
4
x
d
x
? Round to the nearest thousandth if necessary.
\newline
Answer:
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Let
f
f
f
be the function defined by
f
(
x
)
=
3
ln
(
x
)
f(x)=3 \ln (x)
f
(
x
)
=
3
ln
(
x
)
. If six subintervals of equal length are used, what is the value of the left Riemann sum approximation for
∫
1
4
3
ln
(
x
)
d
x
\int_{1}^{4} 3 \ln (x) d x
∫
1
4
3
ln
(
x
)
d
x
? Round to the nearest thousandth if necessary.
\newline
Answer:
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Let
f
f
f
be the function defined by
f
(
x
)
=
3
ln
(
x
)
f(x)=3 \ln (x)
f
(
x
)
=
3
ln
(
x
)
. If three subintervals of equal length are used, what is the value of the trapezoidal sum approximation for
∫
2
3.5
3
ln
(
x
)
d
x
\int_{2}^{3.5} 3 \ln (x) d x
∫
2
3.5
3
ln
(
x
)
d
x
? Round to the nearest thousandth if necessary.
\newline
Answer:
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Let
f
f
f
be the function defined by
f
(
x
)
=
x
2
f(x)=x^{2}
f
(
x
)
=
x
2
. If four subintervals of equal length are used, what is the value of the trapezoidal sum approximation for
∫
2
3
x
2
d
x
\int_{2}^{3} x^{2} d x
∫
2
3
x
2
d
x
? Round to the nearest thousandth if necessary.
\newline
Answer:
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Let
f
f
f
be the function defined by
f
(
x
)
=
4
x
f(x)=4 \sqrt{x}
f
(
x
)
=
4
x
. If four subintervals of equal length are used, what is the value of the trapezoidal sum approximation for
∫
1
7
4
x
d
x
\int_{1}^{7} 4 \sqrt{x} d x
∫
1
7
4
x
d
x
? Round to the nearest thousandth if necessary.
\newline
Answer:
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Let
f
f
f
be the function defined by
f
(
x
)
=
6
ln
(
x
)
f(x)=6 \ln (x)
f
(
x
)
=
6
ln
(
x
)
. If four subintervals of equal length are used, what is the value of the left Riemann sum approximation for
∫
1
9
6
ln
(
x
)
d
x
\int_{1}^{9} 6 \ln (x) d x
∫
1
9
6
ln
(
x
)
d
x
? Round to the nearest thousandth if necessary.
\newline
Answer:
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Let
f
f
f
be the function defined by
f
(
x
)
=
4
x
f(x)=4 \sqrt{x}
f
(
x
)
=
4
x
. If three subintervals of equal length are used, what is the value of the trapezoidal sum approximation for
∫
3
4.5
4
x
d
x
\int_{3}^{4.5} 4 \sqrt{x} d x
∫
3
4.5
4
x
d
x
? Round to the nearest thousandth if necessary.
\newline
Answer:
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Let
f
f
f
be the function defined by
f
(
x
)
=
3
ln
(
x
)
f(x)=3 \ln (x)
f
(
x
)
=
3
ln
(
x
)
. If three subintervals of equal length are used, what is the value of the left Riemann sum approximation for
∫
1
10
3
ln
(
x
)
d
x
\int_{1}^{10} 3 \ln (x) d x
∫
1
10
3
ln
(
x
)
d
x
? Round to the nearest thousandth if necessary.
\newline
Answer:
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Let
f
f
f
be the function defined by
f
(
x
)
=
x
4
f(x)=x^{4}
f
(
x
)
=
x
4
. If five subintervals of equal length are used, what is the value of the midpoint Riemann sum approximation for
∫
1
2
x
4
d
x
\int_{1}^{2} x^{4} d x
∫
1
2
x
4
d
x
? Round to the nearest thousandth if necessary.
\newline
Answer:
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Let
f
f
f
be the function defined by
f
(
x
)
=
5
ln
(
x
)
f(x)=5 \ln (x)
f
(
x
)
=
5
ln
(
x
)
. If three subintervals of equal length are used, what is the value of the trapezoidal sum approximation for
∫
3
7.5
5
ln
(
x
)
d
x
?
\int_{3}^{7.5} 5 \ln (x) d x ?
∫
3
7.5
5
ln
(
x
)
d
x
?
Round to the nearest thousandth if necessary.
\newline
Answer:
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Let
f
f
f
be the function defined by
f
(
x
)
=
4
x
f(x)=4 \sqrt{x}
f
(
x
)
=
4
x
. If four subintervals of equal length are used, what is the value of the midpoint Riemann sum approximation for
∫
2
8
4
x
d
x
\int_{2}^{8} 4 \sqrt{x} d x
∫
2
8
4
x
d
x
? Round to the nearest thousandth if necessary.
\newline
Answer:
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Find the value of
∫
1
5
6
d
x
11
−
2
x
\int_{1}^{5} \frac{6 d x}{11-2 x}
∫
1
5
11
−
2
x
6
d
x
. Express your answer as a constant times
ln
3
\ln 3
ln
3
.
\newline
Answer:
□
ln
3
\square \ln 3
□
ln
3
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Find the value of
∫
3
5
3
d
x
7
−
x
\int_{3}^{5} \frac{3 d x}{7-x}
∫
3
5
7
−
x
3
d
x
. Express your answer as a constant times
ln
2
\ln 2
ln
2
.
\newline
Answer:
□
ln
2
\square \ln 2
□
ln
2
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Find the value of
∫
2
7
5
d
x
x
−
12
\int_{2}^{7} \frac{5 d x}{x-12}
∫
2
7
x
−
12
5
d
x
. Express your answer as a constant times
ln
2
\ln 2
ln
2
.
\newline
Answer:
□
ln
2
\square \ln 2
□
ln
2
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Find the value of
∫
4
6
3
d
x
x
−
8
\int_{4}^{6} \frac{3 d x}{x-8}
∫
4
6
x
−
8
3
d
x
. Express your answer as a constant times
ln
2
\ln 2
ln
2
.
\newline
Answer:
□
ln
2
\square \ln 2
□
ln
2
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Find the value of
∫
4
9
1
3
x
−
11
d
x
\int_{4}^{9} \frac{1}{3 x-11} d x
∫
4
9
3
x
−
11
1
d
x
. Express your answer as a constant times
ln
2
\ln 2
ln
2
.
\newline
Answer:
□
ln
2
\square \ln 2
□
ln
2
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Find the value of
∫
1
4
4
2
x
−
9
d
x
\int_{1}^{4} \frac{4}{2 x-9} d x
∫
1
4
2
x
−
9
4
d
x
. Express your answer as a constant times
ln
7
\ln 7
ln
7
.
\newline
Answer:
□
ln
7
\square \ln 7
□
ln
7
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Find the value of
∫
6
8
4
d
x
4
−
x
\int_{6}^{8} \frac{4 d x}{4-x}
∫
6
8
4
−
x
4
d
x
. Express your answer as a constant times
ln
2
\ln 2
ln
2
.
\newline
Answer:
□
ln
2
\square \ln 2
□
ln
2
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Find the value of
∫
3
8
3
d
x
9
−
x
\int_{3}^{8} \frac{3 d x}{9-x}
∫
3
8
9
−
x
3
d
x
. Express your answer as a constant times
ln
6
\ln 6
ln
6
.
\newline
Answer:
□
ln
6
\square \ln 6
□
ln
6
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Evaluate the integral and express your answer in simplest form.
\newline
∫
−
9
x
x
2
−
25
d
x
\int \frac{-9}{x \sqrt{x^{2}-25}} d x
∫
x
x
2
−
25
−
9
d
x
\newline
Answer:
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Evaluate the integral and express your answer in simplest form.
\newline
∫
−
3
x
x
2
−
1
d
x
\int \frac{-3}{x \sqrt{x^{2}-1}} d x
∫
x
x
2
−
1
−
3
d
x
\newline
Answer:
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Evaluate the integral and express your answer in simplest form.
\newline
∫
9
x
x
2
−
1
d
x
\int \frac{9}{x \sqrt{x^{2}-1}} d x
∫
x
x
2
−
1
9
d
x
\newline
Answer:
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Evaluate the integral and express your answer in simplest form.
\newline
∫
3
x
x
2
−
9
d
x
\int \frac{3}{x \sqrt{x^{2}-9}} d x
∫
x
x
2
−
9
3
d
x
\newline
Answer:
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Evaluate the integral and express your answer in simplest form.
\newline
∫
−
2
x
x
2
−
9
d
x
\int \frac{-2}{x \sqrt{x^{2}-9}} d x
∫
x
x
2
−
9
−
2
d
x
\newline
Answer:
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Evaluate the integral and express your answer in simplest form.
\newline
∫
6
4
−
x
2
d
x
\int \frac{6}{\sqrt{4-x^{2}}} d x
∫
4
−
x
2
6
d
x
\newline
Answer:
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Evaluate
∫
0
4
(
3
e
−
0.25
x
+
8
)
d
x
\int_{0}^{4}\left(3 e^{-0.25 x}+8\right) d x
∫
0
4
(
3
e
−
0.25
x
+
8
)
d
x
and express the answer in simplest form.
\newline
Answer:
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Evaluate
∫
0
6
(
10
e
−
0.5
x
+
4
)
d
x
\int_{0}^{6}\left(10 e^{-0.5 x}+4\right) d x
∫
0
6
(
10
e
−
0.5
x
+
4
)
d
x
and express the answer in simplest form.
\newline
Answer:
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Evaluate the integral and express your answer in simplest form.
\newline
∫
1
x
x
2
−
25
d
x
\int \frac{1}{x \sqrt{x^{2}-25}} d x
∫
x
x
2
−
25
1
d
x
\newline
Answer:
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Evaluate the integral and express your answer in simplest form.
\newline
∫
3
x
x
2
−
36
d
x
\int \frac{3}{x \sqrt{x^{2}-36}} d x
∫
x
x
2
−
36
3
d
x
\newline
Answer:
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∫
cot
x
9
sin
2
x
−
1
d
x
\int \frac{\cot x}{\sqrt{9\sin^{2}x-1}} \, dx
∫
9
s
i
n
2
x
−
1
c
o
t
x
d
x
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Calculate the integral and write the answer in simplest form.
\newline
∫
(
5
x
+
1
)
d
x
\int(5 x+1) d x
∫
(
5
x
+
1
)
d
x
\newline
Answer:
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Calculate the integral and write the answer in simplest form.
\newline
∫
(
−
x
−
3
−
6
)
d
x
\int\left(-x^{-3}-6\right) d x
∫
(
−
x
−
3
−
6
)
d
x
\newline
Answer:
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Evaluate the integral.
\newline
∫
x
(
x
−
4
)
(
2
x
+
5
)
d
x
\int x(x-4)(2 x+5) \mathrm{d} x
∫
x
(
x
−
4
)
(
2
x
+
5
)
d
x
\newline
Answer:
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Evaluate the integral.
\newline
∫
x
2
(
2
x
−
5
)
2
d
x
\int x^{2}(2 x-5)^{2} \mathrm{~d} x
∫
x
2
(
2
x
−
5
)
2
d
x
\newline
Answer:
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Evaluate the integral.
\newline
∫
x
2
(
x
+
5
)
2
d
x
\int x^{2}(x+5)^{2} \mathrm{~d} x
∫
x
2
(
x
+
5
)
2
d
x
\newline
Answer:
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Evaluate the integral.
\newline
∫
(
x
−
1
)
(
4
x
−
5
)
d
x
\int(x-1)(4 x-5) \mathrm{d} x
∫
(
x
−
1
)
(
4
x
−
5
)
d
x
\newline
Answer:
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