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For each of the following functions, find the most general antiderivative. f(x…
Related Topics
Wize University Calculus 1 Textbook > Integrals
Antiderivatives of Trig Functions
4 Activities
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
−
sin
(
x
)
+
x
3
3
+
C
-\sin(x)+\frac{x^3}{3}+C
−
sin
(
x
)
+
3
x
3
+
C
sin
(
x
)
+
x
3
3
+
C
\sin(x)+\frac{x^3}{3}+C
sin
(
x
)
+
3
x
3
+
C
−
sin
(
x
)
−
x
3
3
+
C
-\sin(x)-\frac{x^3}{3}+C
−
sin
(
x
)
−
3
x
3
+
C
sin
(
x
)
−
x
3
3
+
C
\sin(x)-\frac{x^3}{3}+C
sin
(
x
)
−
3
x
3
+
C
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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
.
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: 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
.
Find the most general antiderivative.
f
(
x
)
=
−
cos
x
+
x
2
f(x)=-\cos{x}+x^2
f
(
x
)
=
−
cos
x
+
x
2