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Antiderivatives: Exponential Functions
Related Topics
Wize University Calculus 1 Textbook > Integrals
Antiderivatives & The Indefinite Integrals
5 Activities
Wize University Calculus 1 Textbook > Integrals
Antiderivatives of Exponential Functions
4 Activities
Which of the following is the most general antiderivative of the function
e
3
x
+
7
e^{3x+7}
e
3
x
+
7
? In the functions below,
c
c
c
is an arbitrary constant.
1
7
e
3
x
+
7
+
c
\frac{1}{7}e^{3x + 7} + c
7
1
e
3
x
+
7
+
c
1
3
e
3
x
+
7
+
c
\frac{1}{3}e^{3x + 7} + c
3
1
e
3
x
+
7
+
c
e
3
x
+
7
+
c
e^{3x + 7} + c
e
3
x
+
7
+
c
7
e
3
x
+
7
+
c
7e^{3x + 7} + c
7
e
3
x
+
7
+
c
(
3
x
+
7
)
e
3
x
+
6
+
c
(3x + 7)e^{3x + 6} + c
(
3
x
+
7
)
e
3
x
+
6
+
c
3
e
3
x
+
7
+
c
3e^{3x + 7} + c
3
e
3
x
+
7
+
c
I don't know
Check Submission
More Antiderivatives & The Indefinite Integrals Questions:
Practice: Definite Integral
Practice: Definite Integral
Evaluate
∫
−
2
0
∣
x
2
−
1
∣
d
x
\displaystyle \int_{-2}^0\left|x^2-1\right|dx
∫
−
2
0
x
2
−
1
d
x
.
Antiderivatives: Indefinite integrals
Evaluate the integral
∫
1
+
1
x
+
2
x
2
+
3
x
3
d
x
\displaystyle\int_{ }^{ }1+\frac{1}{x}+\frac{2}{x^2}+\frac{3}{x^3}dx
∫
1
+
x
1
+
x
2
2
+
x
3
3
d
x
.
Antiderivatives: Indefinite integrals
Evaluate the integral
∫
1
+
1
x
+
2
x
2
+
3
x
3
d
x
\displaystyle\int_{ }^{ }1+\frac{1}{x}+\frac{2}{x^2}+\frac{3}{x^3}dx
∫
1
+
x
1
+
x
2
2
+
x
3
3
d
x
.
Antiderivatives: Indefinite Integrals
If
F
(
x
)
=
ln
(
2
x
2
)
F(x) = \ln(2x^2)
F
(
x
)
=
ln
(
2
x
2
)
is an antiderivative of
f
(
x
)
f(x)
f
(
x
)
, find
f
f
f
.
Practice: Definite Integral
Practice: Definite Integral
Evaluate
∫
−
2
0
∣
x
2
−
1
∣
d
x
\displaystyle \int_{-2}^0\left|x^2-1\right|dx
∫
−
2
0
x
2
−
1
d
x
.
Find
∫
(
x
2
3
−
x
5
+
1
x
3
4
)
d
x
\displaystyle \int \left(\sqrt[3]{x^2}-\sqrt[5]{x}+\frac{1}{\sqrt[4]{x^3}}\right)dx
∫
(
3
x
2
−
5
x
+
4
x
3
1
)
d
x
Practice: Indefinite Simplify First
Q:
\textbf{Q:}
Q:
Find
∫
[
x
2
−
4
x
x
+
x
(
2
x
2
−
3
x
)
]
d
x
\displaystyle\int_{ }^{ }\left[\frac{x^2-4\sqrt{x}}{\sqrt{x}}+x\left(2x^2-3x\right)\right]dx
∫
[
x
x
2
−
4
x
+
x
(
2
x
2
−
3
x
)
]
d
x
Practice: Indefinite Integral
Evaluate the integral
∫
x
+
1
+
1
x
+
2
x
2
d
x
\displaystyle\int_{ }^{ }x+1+\frac{1}{x}+\frac{2}{x^2}dx
∫
x
+
1
+
x
1
+
x
2
2
d
x
Practice: Indefinite Integral
Evaluate the integral
∫
1
+
1
x
+
2
x
2
+
3
x
3
d
x
\displaystyle\int_{ }^{ }1+\frac{1}{x}+\frac{2}{x^2}+\frac{3}{x^3}dx
∫
1
+
x
1
+
x
2
2
+
x
3
3
d
x
.
Antiderivatives and Indefinite Integrals
Find the most general antiderivative of
f
(
x
)
=
e
2
f\left(x\right)=e^2
f
(
x
)
=
e
2
Riemann Sums
Evaluate
lim
n
→
∞
∑
i
=
1
n
[
5
n
[
(
1
+
5
i
n
)
2
−
3
]
]
\displaystyle \lim_{n\to\infty}\ \sum_{i=1}^n\left[\frac{5}{n}\left[\left(1+\frac{5i}{n}\right)^2-3\right]\right]
n
→
∞
lim
i
=
1
∑
n
[
n
5
[
(
1
+
n
5
i
)
2
−
3
]
]
Find
∫
x
e
3
d
x
\int_{ }^{ }xe^3\ dx
∫
x
e
3
d
x
Antiderivatives: Indefinite Integrals
If
F
(
x
)
=
ln
(
2
x
2
)
F(x) = \ln(2x^2)
F
(
x
)
=
ln
(
2
x
2
)
is an antiderivative of
f
(
x
)
f(x)
f
(
x
)
, find
f
f
f
.
Antiderivatives: Indefinite Integrals
The graph of an
antiderivative
of
f
(
x
)
f(x)
f
(
x
)
is given below. Graph
f
(
x
)
f(x)
f
(
x
)
.
Find an antiderivative for
f
(
x
)
=
x
5
+
2
x
4
x
\displaystyle f(x)=\frac{x^5+2\sqrt{x}}{4x}
f
(
x
)
=
4
x
x
5
+
2
x
Find an antiderivative for
f
(
x
)
=
x
5
+
2
x
4
x
\displaystyle f(x)=\frac{x^5+2\sqrt{x}}{4x}
f
(
x
)
=
4
x
x
5
+
2
x
Find
∫
l
n
(
e
4
x
)
d
x
\displaystyle \int ln(e^{4x})dx
∫
l
n
(
e
4
x
)
d
x
If
F
(
x
)
=
ln
(
2
x
2
)
F(x)=\ln(2x^2)
F
(
x
)
=
ln
(
2
x
2
)
is an antiderivative of
f
(
x
)
f(x)
f
(
x
)
, find
f
.
Antiderivatives: Indefinite integrals
Evaluate the integral
∫
1
+
1
x
+
2
x
2
+
3
x
3
d
x
\displaystyle\int_{ }^{ }1+\frac{1}{x}+\frac{2}{x^2}+\frac{3}{x^3}dx
∫
1
+
x
1
+
x
2
2
+
x
3
3
d
x
.
Practice: Indefinite Simplify First
Q:
\textbf{Q:}
Q:
Find
∫
[
x
2
−
4
x
x
+
x
(
2
x
2
−
3
x
)
]
d
x
\displaystyle\int_{ }^{ }\left[\frac{x^2-4\sqrt{x}}{\sqrt{x}}+x\left(2x^2-3x\right)\right]dx
∫
[
x
x
2
−
4
x
+
x
(
2
x
2
−
3
x
)
]
d
x
Find
∫
(
x
2
3
−
x
5
+
1
x
3
4
)
d
x
\displaystyle \int \left(\sqrt[3]{x^2}-\sqrt[5]{x}+\frac{1}{\sqrt[4]{x^3}}\right)dx
∫
(
3
x
2
−
5
x
+
4
x
3
1
)
d
x
Antiderivatives: Indefinite integrals
Evaluate the indefinite integral
∫
x
2
3
−
x
5
+
1
x
3
4
d
x
\displaystyle \int \sqrt[3]{x^2}-\sqrt[5]{x}+\frac{1}{\sqrt[4]{x^3}}dx
∫
3
x
2
−
5
x
+
4
x
3
1
d
x
.
Indefinite Integrals: Antiderivatives
Find the indefinite integral
∫
2
x
(
2
x
+
3
)
2
d
x
\int2x(2x+3)^2dx
∫
2
x
(
2
x
+
3
)
2
d
x
Find
∫
x
e
3
d
x
\int_{ }^{ }xe^3\ dx
∫
x
e
3
d
x
Antiderivatives: Indefinite Integrals
Find
∫
[
x
2
−
4
x
x
+
x
(
2
x
2
−
3
x
)
]
d
x
\int_{ }^{ }\left[\frac{x^2-4\sqrt{x}}{\sqrt{x}}+x\left(2x^2-3x\right)\right]dx
∫
[
x
x
2
−
4
x
+
x
(
2
x
2
−
3
x
)
]
d
x
.
Practice: Definite Integral
Practice: Definite Integral
Evaluate
∫
−
2
0
∣
x
2
−
1
∣
d
x
\displaystyle \int_{-2}^0\left|x^2-1\right|dx
∫
−
2
0
x
2
−
1
d
x
.
Antiderivatives: Indefinite Integrals
For the following function, find the most general antiderivative.
f
(
x
)
=
x
5
+
2
x
4
x
\displaystyle f(x)=\frac{x^5+2\sqrt{x}}{4x}
f
(
x
)
=
4
x
x
5
+
2
x
Find the antiderivative
F
F
F
f
(
x
)
=
4
x
+
x
2
2
+
5
,
F
(
1
)
=
1
/
3
\displaystyle f(x)=4x+\frac{x^2}{2}+5, F(1)=1/3
f
(
x
)
=
4
x
+
2
x
2
+
5
,
F
(
1
)
=
1/3
More Antiderivatives of Exponential Functions Questions:
Evaluate the indefinite integral
∫
6
x
+
e
−
6
x
−
3
x
2
+
1
d
x
\displaystyle \int6^x+e^{-6x}-\frac{3}{x^2+1}dx
∫
6
x
+
e
−
6
x
−
x
2
+
1
3
d
x
.
Evaluate the indefinite integral
∫
6
x
+
e
−
6
x
−
3
x
2
+
1
d
x
\displaystyle \int6^x+e^{-6x}-\frac{3}{x^2+1}dx
∫
6
x
+
e
−
6
x
−
x
2
+
1
3
d
x
.
Practice: Definite Integral (~F2017 Final Q26)
Practice: Definite Integral
Evaluate
∫
0
3
1
9
+
x
2
+
2
x
ln
2
d
x
\int_0^3\frac{1}{9+x^2}+2^x\ln2\ dx
∫
0
3
9
+
x
2
1
+
2
x
ln
2
d
x
.
Antiderivatives: Exponential Functions
Find the indefinite integral
∫
(
6
x
+
e
−
6
x
)
d
x
\displaystyle \int(6^x+e^{-6x}) \ dx
∫
(
6
x
+
e
−
6
x
)
d
x
Exponential Functions: $f$ from $f''$
Given
f
′
′
(
x
)
=
5
e
x
+
2
sin
x
f''(x)=5e^x+2\sin x
f
′′
(
x
)
=
5
e
x
+
2
sin
x
and given
f
(
0
)
=
0
f(0)=0
f
(
0
)
=
0
,
f
(
π
)
=
0
f(\pi)=0
f
(
π
)
=
0
, find
f
(
x
)
f(x)
f
(
x
)
. Enter "f(x)=..."
Indefinite Integrals with Exponential
Compute the following integral:
∫
3
−
x
d
x
\displaystyle \int 3^{-x}\,\text{d}x
∫
3
−
x
d
x
.
Use upper case
C
C
C
to denote any constants.
Find
∫
(
c
o
s
7
)
e
3
x
d
x
\displaystyle \int (cos7)e^{3x}dx
∫
(
cos
7
)
e
3
x
d
x
Antiderivatives: Exponential Functions
Find an antiderivative of
f
(
x
)
=
2
e
3
x
−
2
f(x) = 2e^{3x-2}
f
(
x
)
=
2
e
3
x
−
2
Practice: Definite Integral
Evaluate
∫
0
3
1
9
+
x
2
+
2
x
ln
2
d
x
\displaystyle \int_0^3\frac{1}{9+x^2}+2^x\ln2\ dx
∫
0
3
9
+
x
2
1
+
2
x
ln
2
d
x
.
Find the equation of the tangent line of the graph
y
=
e
x
cos
x
y=\frac{e^x}{\cos\ x}
y
=
c
o
s
x
e
x
at the point where 𝑥 = 0.
Exponential Functions: $f$ from $f''$
Given
f
′
′
(
x
)
=
5
e
x
+
2
sin
x
f''(x)=5e^x+2\sin x
f
′′
(
x
)
=
5
e
x
+
2
sin
x
and given
f
(
0
)
=
0
f(0)=0
f
(
0
)
=
0
,
f
(
π
)
=
0
f(\pi)=0
f
(
π
)
=
0
, find
f
(
x
)
f(x)
f
(
x
)
. Enter "f(x)=..."
Indefinite Integrals with Exponential
Compute the following integral:
∫
3
−
x
d
x
\displaystyle \int 3^{-x}\,\text{d}x
∫
3
−
x
d
x
.
Use upper case
C
C
C
to denote any constants.
Antiderivatives: Exponential Functions
Find the indefinite integral
∫
(
6
x
+
e
−
6
x
)
d
x
\displaystyle \int(6^x+e^{-6x}) \ dx
∫
(
6
x
+
e
−
6
x
)
d
x
Find the equation of the tangent line to the curve
y
=
2
3
e
3
−
3
x
y=2^3e^{3-3x}
y
=
2
3
e
3
−
3
x
at
x
=
1
x=1
x
=
1
.
Enter your answer as y=...
Evaluate the indefinite integral
∫
6
x
+
e
−
6
x
−
3
x
2
+
1
d
x
\displaystyle \int6^x+e^{-6x}-\frac{3}{x^2+1}dx
∫
6
x
+
e
−
6
x
−
x
2
+
1
3
d
x
.
Higher Order Derivatives
Find a general formula for the
n
th-derivative of
f
(
x
)
=
e
3
x
f(x)=e^{3x}
f
(
x
)
=
e
3
x
.
Antiderivatives: Trigonometric and Exponential Functions
If
f
′
′
(
x
)
=
cos
(
2
x
)
−
x
4
+
e
x
f''\left(x\right)=\cos\left(2x\right)-x^4+e^x
f
′′
(
x
)
=
cos
(
2
x
)
−
x
4
+
e
x
,
f
(
0
)
=
0
=
f
′
(
0
)
f\left(0\right)=0=f'\left(0\right)
f
(
0
)
=
0
=
f
′
(
0
)
Practice: Definite Integral
Practice: Definite Integral
Evaluate
∫
0
3
1
9
+
x
2
+
2
x
ln
2
d
x
\int_0^3\frac{1}{9+x^2}+2^x\ln2\ dx
∫
0
3
9
+
x
2
1
+
2
x
ln
2
d
x
.