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Antiderivatives: Trigonometric and Exponential Functions
Related Topics
Wize University Calculus 1 Textbook > Integrals
Antiderivatives of Trig Functions
4 Activities
Wize University Calculus 1 Textbook > Integrals
Antiderivatives of Exponential Functions
4 Activities
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
)
f
(
x
)
=
cos
2
x
4
−
x
6
30
+
e
x
−
x
+
7
2
f\left(x\right)=\frac{\cos2x}{4}-\frac{x^6}{30}+e^x-x+\frac{7}{2}
f
(
x
)
=
4
c
o
s
2
x
−
30
x
6
+
e
x
−
x
+
2
7
f
(
x
)
=
cos
2
x
4
−
x
6
30
+
e
x
−
x
+
7
2
f\left(x\right)=\frac{\cos2x}{4}-\frac{x^6}{30}+e^x-x+\frac{7}{2}
f
(
x
)
=
4
c
o
s
2
x
−
30
x
6
+
e
x
−
x
+
2
7
f
(
x
)
=
−
cos
2
x
4
−
x
6
30
+
e
x
−
x
−
3
4
f\left(x\right)=-\frac{\cos2x}{4}-\frac{x^6}{30}+e^x-x-\frac{3}{4}
f
(
x
)
=
−
4
c
o
s
2
x
−
30
x
6
+
e
x
−
x
−
4
3
f
(
x
)
=
−
cos
2
x
4
−
x
6
30
+
e
x
−
x
+
7
2
f\left(x\right)=-\frac{\cos2x}{4}-\frac{x^6}{30}+e^x-x+\frac{7}{2}
f
(
x
)
=
−
4
c
o
s
2
x
−
30
x
6
+
e
x
−
x
+
2
7
f
(
x
)
=
−
cos
2
x
4
−
x
6
30
+
e
x
−
x
+
5
2
f\left(x\right)=-\frac{\cos2x}{4}-\frac{x^6}{30}+e^x-x+\frac{5}{2}
f
(
x
)
=
−
4
c
o
s
2
x
−
30
x
6
+
e
x
−
x
+
2
5
I don't know
Check Submission
More Antiderivatives of Trig Functions Questions:
Derivatives/Antiderivatives
What is the derivative of
f
(
x
)
=
e
x
cos
x
f(x)=e^{x}\cos x
f
(
x
)
=
e
x
cos
x
?
Derivatives/Antiderivatives
What is the derivative of
f
(
x
)
=
e
2
x
cos
3
x
f(x)=e^{2x}\cos3x
f
(
x
)
=
e
2
x
cos
3
x
?
Indefinite Integrals with Trig and Inverse Trig
Find
∫
(
2
x
−
3
4
−
x
2
+
2
sin
x
)
d
x
\displaystyle\int_{ }^{ }\left(2\sqrt{x}-\frac{3}{\sqrt{4-x^2}}+2\sin x\right)dx
∫
(
2
x
−
4
−
x
2
3
+
2
sin
x
)
d
x
Practice: Indefinite Integral
Find
f
(
x
)
f\left(x\right)
f
(
x
)
, given that
f
′
(
x
)
=
x
2
−
2
+
sin
x
f'\left(x\right)=x^2-2+\sin x
f
′
(
x
)
=
x
2
−
2
+
sin
x
and
f
(
0
)
=
3
f\left(0\right)=3
f
(
0
)
=
3
.
Practice: Indefinite Integral
Practice: Indefinite Integral
Find
f
(
x
)
f\left(x\right)
f
(
x
)
, given that
f
′
(
x
)
=
x
2
−
2
+
sin
x
f'\left(x\right)=x^2-2+\sin x
f
′
(
x
)
=
x
2
−
2
+
sin
x
and
f
(
0
)
=
3
f\left(0\right)=3
f
(
0
)
=
3
.
Indefinite Integrals with Trig and Inverse Trig
Find
∫
(
2
x
−
3
4
−
x
2
+
2
sin
x
)
d
x
\displaystyle\int_{ }^{ }\left(2\sqrt{x}-\frac{3}{\sqrt{4-x^2}}+2\sin x\right)dx
∫
(
2
x
−
4
−
x
2
3
+
2
sin
x
)
d
x
Antiderivatives: Trigonometric Functions
Evaluate the integral
∫
[
tan
2
x
+
1
+
1
sec
2
x
+
sin
2
x
]
d
x
\displaystyle \int_{ }^{ }\left[\tan^2x+1+\frac{1}{\sec^2x}+\sin^2x\right]dx
∫
[
tan
2
x
+
1
+
sec
2
x
1
+
sin
2
x
]
d
x
.
Hint: First simplify using trig identities
Find the value of the definite integral
∫
π
/
6
π
/
3
(
sec
x
+
tan
x
)
sec
x
d
x
\displaystyle \int_{\pi/6 }^{\pi/3}(\sec x+\tan x)\sec x dx
∫
π
/6
π
/3
(
sec
x
+
tan
x
)
sec
x
d
x
.
Evaluate the indefinite integral
∫
sec
θ
tan
θ
−
sin
(
θ
5
)
d
θ
\displaystyle \int\sec\theta\tan\theta-\sin\left(\frac{\theta}{5}\right)d\theta
∫
sec
θ
tan
θ
−
sin
(
5
θ
)
d
θ
.
Evaluate the indefinite integral
∫
sec
θ
tan
θ
−
sin
(
θ
5
)
d
θ
\displaystyle \int\sec\theta\tan\theta-\sin\left(\frac{\theta}{5}\right)d\theta
∫
sec
θ
tan
θ
−
sin
(
5
θ
)
d
θ
.
Practice: Indefinite Integral (~F2018 Final Q26)
Practice: Indefinite Integral
Find
f
(
x
)
f\left(x\right)
f
(
x
)
, given that
f
′
(
x
)
=
x
2
−
2
+
sin
x
f'\left(x\right)=x^2-2+\sin x
f
′
(
x
)
=
x
2
−
2
+
sin
x
and
f
(
0
)
=
3
f\left(0\right)=3
f
(
0
)
=
3
.
Indefinite Integrals with Trig and Inverse Trig
Q:
\textbf{Q:}
Q:
Find
∫
(
2
x
−
3
4
−
x
2
+
2
sin
x
)
d
x
\displaystyle\int_{ }^{ }\left(2\sqrt{x}-\frac{3}{\sqrt{4-x^2}}+2\sin x\right)dx
∫
(
2
x
−
4
−
x
2
3
+
2
sin
x
)
d
x
Antiderivatives: Indefinite with Trig
Find
∫
(
tan
2
x
+
1
sec
2
x
+
1
+
x
π
+
sin
2
x
)
d
x
\displaystyle\int_{ }^{ }\left(\tan^2x+\frac{1}{\sec^2x}+1+x^\pi+\sin^2x\right)dx
∫
(
tan
2
x
+
sec
2
x
1
+
1
+
x
π
+
sin
2
x
)
d
x
Antiderivatives: Product with Trig
Find
d
f
d
θ
\displaystyle\frac{df}{d\theta}
d
θ
df
given
f
(
θ
)
=
(
sin
θ
−
cos
θ
)
sec
θ
\displaystyle f(\theta)=(\sin\theta-\cos\theta)\sec\theta
f
(
θ
)
=
(
sin
θ
−
cos
θ
)
sec
θ
Practice: Indefinite Integral
Evaluate the integral
∫
[
tan
2
x
+
1
+
1
sec
2
x
+
sin
2
x
]
d
x
\displaystyle \int_{ }^{ }\left[\tan^2x+1+\frac{1}{\sec^2x}+\sin^2x\right]dx
∫
[
tan
2
x
+
1
+
sec
2
x
1
+
sin
2
x
]
d
x
.
Hint: First simplify using trig identities
Practice: Indefinite Integral
Evaluate the integral
∫
[
tan
2
x
+
1
+
1
sec
2
x
+
sin
2
x
]
d
x
\displaystyle \int_{ }^{ }\left[\tan^2x+1+\frac{1}{\sec^2x}+\sin^2x\right]dx
∫
[
tan
2
x
+
1
+
sec
2
x
1
+
sin
2
x
]
d
x
.
Hint: First simplify using trig identities
Find
∫
0
π
∣
cos
x
∣
d
x
\displaystyle\int_0^{\pi}\left|\cos\ x\right|dx
∫
0
π
∣
cos
x
∣
d
x
Antiderivatives: Trigonometric Functions
Find
∫
1
+
sin
3
x
sin
2
x
d
x
\int \frac{1+\sin^3 x}{\sin^2 x} dx
∫
sin
2
x
1
+
sin
3
x
d
x
Find an antiderivative for
f
(
x
)
=
−
cos
x
+
x
2
f(x)=-\cos{x}+x^2
f
(
x
)
=
−
cos
x
+
x
2
Antiderivatives: Trigonometric Functions
Find the definite integral
∫
π
/
6
π
/
3
(
sec
x
+
tan
x
)
sec
x
d
x
\int_{\pi/6 }^{\pi/3}(\sec x+\tan x)\sec x dx
∫
π
/6
π
/3
(
sec
x
+
tan
x
)
sec
x
d
x
final114
Find the indefinite integral
∫
(
1
x
2
+
sin
x
)
d
x
\displaystyle\int\bigg(\frac{1}{x^2}+\sin x\bigg)dx
∫
(
x
2
1
+
sin
x
)
d
x
Antiderivatives: Trigonometric Functions
Evaluate the integral
∫
[
tan
2
x
+
1
+
1
sec
2
x
+
sin
2
x
]
d
x
\displaystyle \int_{ }^{ }\left[\tan^2x+1+\frac{1}{\sec^2x}+\sin^2x\right]dx
∫
[
tan
2
x
+
1
+
sec
2
x
1
+
sin
2
x
]
d
x
.
Hint: First simplify using trig identities
final114
Find the definite integral
∫
π
/
6
π
/
3
(
sec
x
+
tan
x
)
sec
x
d
x
\int_{\pi/6 }^{\pi/3}(\sec x+\tan x)\sec x dx
∫
π
/6
π
/3
(
sec
x
+
tan
x
)
sec
x
d
x
Find the exact value of
∫
0
π
sin
x
d
x
\int_0^{\pi}\sin xdx
∫
0
π
sin
x
d
x
Antiderivatives: Product with Trig
Find
d
f
d
θ
\displaystyle\frac{df}{d\theta}
d
θ
df
given
f
(
θ
)
=
(
sin
θ
−
cos
θ
)
sec
θ
\displaystyle f(\theta)=(\sin\theta-\cos\theta)\sec\theta
f
(
θ
)
=
(
sin
θ
−
cos
θ
)
sec
θ
Find the indefinite integral
∫
(
1
x
2
+
sin
x
)
d
x
\displaystyle\int\bigg(\frac{1}{x^2}+\sin x\bigg)dx
∫
(
x
2
1
+
sin
x
)
d
x
. Use C for the integration constant.
Antiderivatives: Indefinite with Trig
Find
∫
(
tan
2
x
+
1
sec
2
x
+
1
+
x
π
+
sin
2
x
)
d
x
\displaystyle\int_{ }^{ }\left(\tan^2x+\frac{1}{\sec^2x}+1+x^\pi+\sin^2x\right)dx
∫
(
tan
2
x
+
sec
2
x
1
+
1
+
x
π
+
sin
2
x
)
d
x
Indefinite Integrals with Trig and Inverse Trig
Find
∫
(
2
x
−
3
4
−
x
2
+
2
sin
x
)
d
x
\displaystyle\int_{ }^{ }\left(2\sqrt{x}-\frac{3}{\sqrt{4-x^2}}+2\sin x\right)dx
∫
(
2
x
−
4
−
x
2
3
+
2
sin
x
)
d
x
Derivatives/Antiderivatives
Compute the second derivative of
f
(
x
)
=
e
tan
x
f(x)=e^{\tan x}
f
(
x
)
=
e
t
a
n
x
.
Find the value of the definite integral
∫
π
/
6
π
/
3
(
sec
x
+
tan
x
)
sec
x
d
x
\displaystyle \int_{\pi/6 }^{\pi/3}(\sec x+\tan x)\sec x dx
∫
π
/6
π
/3
(
sec
x
+
tan
x
)
sec
x
d
x
.
Evaluate the indefinite integral
∫
sec
θ
tan
θ
−
sin
(
θ
5
)
d
θ
\displaystyle \int\sec\theta\tan\theta-\sin\left(\frac{\theta}{5}\right)d\theta
∫
sec
θ
tan
θ
−
sin
(
5
θ
)
d
θ
.
Find
∫
0
π
∣
cos
x
∣
d
x
\displaystyle\int_0^{\pi}\left|\cos\ x\right|dx
∫
0
π
∣
cos
x
∣
d
x
Antiderivatives: Trigonometric Functions
Evaluate
∫
π
3
π
∣
cos
x
∣
d
x
\int_{\frac{\pi}{3}}^{\pi}\left|\cos x\right|dx
∫
3
π
π
∣
cos
x
∣
d
x
.
Practice: Indefinite Integral
Practice: Indefinite Integral
Find
f
(
x
)
f\left(x\right)
f
(
x
)
, given that
f
′
(
x
)
=
x
2
−
2
+
sin
x
f'\left(x\right)=x^2-2+\sin x
f
′
(
x
)
=
x
2
−
2
+
sin
x
and
f
(
0
)
=
3
f\left(0\right)=3
f
(
0
)
=
3
.
For each of the following functions, find the most general antiderivative.
f
(
x
)
=
−
cos
x
+
x
2
f(x)=-\cos{x}+x^2
f
(
x
)
=
−
cos
x
+
x
2
Find the most general antiderivative.
f
(
x
)
=
−
cos
x
+
x
2
f(x)=-\cos{x}+x^2
f
(
x
)
=
−
cos
x
+
x
2
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.
Antiderivatives: Exponential Functions
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.
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
.
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
.