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Practice: Indefinite Integral

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Wize University Calculus 2 Textbook > Integration

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Practice: Indefinite Integral

Find f(x)f\left(x\right), given that f′(x)=x2−2+sin⁡xf'\left(x\right)=x^2-2+\sin x and f(0)=3f\left(0\right)=3.

More Indefinite Integrals Questions:
Practice: Indefinite Integral
Find f(x)f\left(x\right), given that f′(x)=x2−2+sin⁡xf'\left(x\right)=x^2-2+\sin x and f(0)=3f\left(0\right)=3.
Practice: Indefinite Integral
Practice: Indefinite Integral
Find f(x)f\left(x\right), given that f′(x)=x2−2+sin⁡xf'\left(x\right)=x^2-2+\sin x and f(0)=3f\left(0\right)=3.
Indefinite Integrals: Trigonometric Functions
Evaluate the following indefinite integral
∫(sec⁡2x)(cos⁡22x)dx\displaystyle \int (\sec 2x)( \cos^2 2x)dx
Antiderivatives and Indefinite Integrals
Find the most general antiderivative of f(x)=e2f\left(x\right)=e^2
Indefinite Integrals with Exponential
Compute the following integral: ∫3−x dx\displaystyle \int 3^{-x}\,\text{d}x.
Use upper case CC to denote any constants.
Antiderivatives: Indefinite integrals
Evaluate the integral ∫1+1x+2x2+3x3dx\displaystyle\int_{ }^{ }1+\frac{1}{x}+\frac{2}{x^2}+\frac{3}{x^3}dx.
Indefinite Integrals
Evaluate the following indefinite integral
∫x4−3xxdx\displaystyle \int \frac{x^4-3x}{x}dx
Antiderivatives: Indefinite integrals
Evaluate the indefinite integral ∫x23−x5+1x34dx\displaystyle \int \sqrt[3]{x^2}-\sqrt[5]{x}+\frac{1}{\sqrt[4]{x^3}}dx.
More Antiderivatives of Trig Functions Questions:
Derivatives/Antiderivatives
What is the derivative of f(x)=excos⁡xf(x)=e^{x}\cos x?
Derivatives/Antiderivatives
What is the derivative of f(x)=e2xcos⁡3xf(x)=e^{2x}\cos3x?
Indefinite Integrals with Trig and Inverse Trig
Find ∫(2x−34−x2+2sin⁡x)dx\displaystyle\int_{ }^{ }\left(2\sqrt{x}-\frac{3}{\sqrt{4-x^2}}+2\sin x\right)dx
Practice: Indefinite Integral
Find f(x)f\left(x\right), given that f′(x)=x2−2+sin⁡xf'\left(x\right)=x^2-2+\sin x and f(0)=3f\left(0\right)=3.
Practice: Indefinite Integral
Practice: Indefinite Integral
Find f(x)f\left(x\right), given that f′(x)=x2−2+sin⁡xf'\left(x\right)=x^2-2+\sin x and f(0)=3f\left(0\right)=3.
Indefinite Integrals with Trig and Inverse Trig
Find ∫(2x−34−x2+2sin⁡x)dx\displaystyle\int_{ }^{ }\left(2\sqrt{x}-\frac{3}{\sqrt{4-x^2}}+2\sin x\right)dx
Antiderivatives: Trigonometric Functions
Evaluate the integral∫[tan⁡2x+1+1sec⁡2x+sin⁡2x]dx\displaystyle \int_{ }^{ }\left[\tan^2x+1+\frac{1}{\sec^2x}+\sin^2x\right]dx.
Hint: First simplify using trig identities
Find the value of the definite integral ∫π/6π/3(sec⁡x+tan⁡x)sec⁡xdx\displaystyle \int_{\pi/6 }^{\pi/3}(\sec x+\tan x)\sec x dx .
Evaluate the indefinite integral ∫sec⁡θtan⁡θ−sin⁡(θ5)dθ\displaystyle \int\sec\theta\tan\theta-\sin\left(\frac{\theta}{5}\right)d\theta.
Evaluate the indefinite integral ∫sec⁡θtan⁡θ−sin⁡(θ5)dθ\displaystyle \int\sec\theta\tan\theta-\sin\left(\frac{\theta}{5}\right)d\theta.
Practice: Indefinite Integral (~F2018 Final Q26)
Practice: Indefinite Integral
Find f(x)f\left(x\right), given that f′(x)=x2−2+sin⁡xf'\left(x\right)=x^2-2+\sin x and f(0)=3f\left(0\right)=3.
Indefinite Integrals with Trig and Inverse Trig
Q:\textbf{Q:} Find ∫(2x−34−x2+2sin⁡x)dx\displaystyle\int_{ }^{ }\left(2\sqrt{x}-\frac{3}{\sqrt{4-x^2}}+2\sin x\right)dx
Antiderivatives: Indefinite with Trig
Find ∫(tan⁡2x+1sec⁡2x+1+xπ+sin⁡2x)dx\displaystyle\int_{ }^{ }\left(\tan^2x+\frac{1}{\sec^2x}+1+x^\pi+\sin^2x\right)dx
Antiderivatives: Product with Trig
Find dfdθ\displaystyle\frac{df}{d\theta} given f(θ)=(sin⁡θ−cos⁡θ)sec⁡θ\displaystyle f(\theta)=(\sin\theta-\cos\theta)\sec\theta
Practice: Indefinite Integral
Evaluate the integral∫[tan⁡2x+1+1sec⁡2x+sin⁡2x]dx\displaystyle \int_{ }^{ }\left[\tan^2x+1+\frac{1}{\sec^2x}+\sin^2x\right]dx.
Hint: First simplify using trig identities
Practice: Indefinite Integral
Evaluate the integral∫[tan⁡2x+1+1sec⁡2x+sin⁡2x]dx\displaystyle \int_{ }^{ }\left[\tan^2x+1+\frac{1}{\sec^2x}+\sin^2x\right]dx.
Hint: First simplify using trig identities
Find ∫0π∣cos⁡ x∣dx\displaystyle\int_0^{\pi}\left|\cos\ x\right|dx
Antiderivatives: Trigonometric Functions
Find
∫1+sin⁡3xsin⁡2xdx\int \frac{1+\sin^3 x}{\sin^2 x} dx
Find an antiderivative for
f(x)=−cos⁡x+x2f(x)=-\cos{x}+x^2
Antiderivatives: Trigonometric Functions
Find the definite integral ∫π/6π/3(sec⁡x+tan⁡x)sec⁡xdx\int_{\pi/6 }^{\pi/3}(\sec x+\tan x)\sec x dx
final114
Find the indefinite integral ∫(1x2+sin⁡x)dx\displaystyle\int\bigg(\frac{1}{x^2}+\sin x\bigg)dx
Antiderivatives: Trigonometric Functions
Evaluate the integral∫[tan⁡2x+1+1sec⁡2x+sin⁡2x]dx\displaystyle \int_{ }^{ }\left[\tan^2x+1+\frac{1}{\sec^2x}+\sin^2x\right]dx.
Hint: First simplify using trig identities
final114
Find the definite integral ∫π/6π/3(sec⁡x+tan⁡x)sec⁡xdx\int_{\pi/6 }^{\pi/3}(\sec x+\tan x)\sec x dx
Find the exact value of
∫0πsin⁡xdx\int_0^{\pi}\sin xdx
Antiderivatives: Product with Trig
Find dfdθ\displaystyle\frac{df}{d\theta} given f(θ)=(sin⁡θ−cos⁡θ)sec⁡θ\displaystyle f(\theta)=(\sin\theta-\cos\theta)\sec\theta
Find the indefinite integral ∫(1x2+sin⁡x)dx\displaystyle\int\bigg(\frac{1}{x^2}+\sin x\bigg)dx . Use C for the integration constant.
Antiderivatives: Indefinite with Trig
Find ∫(tan⁡2x+1sec⁡2x+1+xπ+sin⁡2x)dx\displaystyle\int_{ }^{ }\left(\tan^2x+\frac{1}{\sec^2x}+1+x^\pi+\sin^2x\right)dx
Indefinite Integrals with Trig and Inverse Trig
Find ∫(2x−34−x2+2sin⁡x)dx\displaystyle\int_{ }^{ }\left(2\sqrt{x}-\frac{3}{\sqrt{4-x^2}}+2\sin x\right)dx
Derivatives/Antiderivatives
Compute the second derivative of f(x)=etan⁡xf(x)=e^{\tan x}.
Find the value of the definite integral ∫π/6π/3(sec⁡x+tan⁡x)sec⁡xdx\displaystyle \int_{\pi/6 }^{\pi/3}(\sec x+\tan x)\sec x dx .
Evaluate the indefinite integral ∫sec⁡θtan⁡θ−sin⁡(θ5)dθ\displaystyle \int\sec\theta\tan\theta-\sin\left(\frac{\theta}{5}\right)d\theta.
Find ∫0π∣cos⁡ x∣dx\displaystyle\int_0^{\pi}\left|\cos\ x\right|dx
Antiderivatives: Trigonometric Functions
Evaluate ∫π3π∣cos⁡x∣dx\int_{\frac{\pi}{3}}^{\pi}\left|\cos x\right|dx.
Antiderivatives: Trigonometric and Exponential Functions
If f′′(x)=cos⁡(2x)−x4+exf''\left(x\right)=\cos\left(2x\right)-x^4+e^x, f(0)=0=f′(0)f\left(0\right)=0=f'\left(0\right)
For each of the following functions, find the most general antiderivative.
f(x)=−cos⁡x+x2f(x)=-\cos{x}+x^2
Find the most general antiderivative.
f(x)=−cos⁡x+x2f(x)=-\cos{x}+x^2
More Initial Conditions Questions:
Indefinite Integral
Find f(x)f\left(x\right), given that f′(x)=x2−2+sin⁡xf'\left(x\right)=x^2-2+\sin x and f(0)=3f\left(0\right)=3.
Applications of Integration: Position, Velocity and Acceleration
If the accelertion of a particle is given by a(x)=2x+xa\left(x\right)=2^x+x, and v(0)=s(0)=0v\left(0\right)=s\left(0\right)=0, then the displacement of the particle s(x)s\left(x\right) is
Practice: Indefinite Integral
Find f(x)f\left(x\right), given that f′(x)=x2−2+sin⁡xf'\left(x\right)=x^2-2+\sin x and f(0)=3f\left(0\right)=3.
Practice: Indefinite Integral
Practice: Indefinite Integral
Find f(x)f\left(x\right), given that f′(x)=x2−2+sin⁡xf'\left(x\right)=x^2-2+\sin x and f(0)=3f\left(0\right)=3.
Indefinite Integral
Find f(x)f\left(x\right), given that f′(x)=x2−2+sin⁡xf'\left(x\right)=x^2-2+\sin x and f(0)=3f\left(0\right)=3.
Practice: Indefinite Integral (~F2018 Final Q26)
Practice: Indefinite Integral
Find f(x)f\left(x\right), given that f′(x)=x2−2+sin⁡xf'\left(x\right)=x^2-2+\sin x and f(0)=3f\left(0\right)=3.
Exponential Functions: $f$ from $f''$
Given f′′(x)=5ex+2sin⁡xf''(x)=5e^x+2\sin x and given f(0)=0f(0)=0, f(π)=0f(\pi)=0, find f(x)f(x). Enter "f(x)=..."
Practice: Indefinite Integral
Find f(x)f\left(x\right), given that f′′(x)=x2−2+sin⁡xf''\left(x\right)=x^2-2+\sin x, f(0)=3f\left(0\right)=3 and f′(3)=1f'(3)=1
Practice: Indefinite Integral
Find f(x)f\left(x\right), given that f′(x)=x2−2+sin⁡xf'\left(x\right)=x^2-2+\sin x and f(0)=3f\left(0\right)=3.
Applications of Integration: Position, Velocity and Acceleration
If the acceleartion of a particle is given by a(x)=2x+xa\left(x\right)=2^x+x, and v(0)=s(0)=0v\left(0\right)=s\left(0\right)=0, then the displacement of the particle s(x)s\left(x\right) is
Initial Conditions
Find the function y=f(x)y=f(x) if
y′′=3x−5/2y′(1)=1y(4)=6y''=3x^{-5/2} \quad y'(1)=1 \quad y(4)=6
Initial Conditions
Solve the initial value problem
y′=x2+1xy'=\frac{x^2+1}{\sqrt{x}}
y(1)=3y(1)=3
Initial Conditions
Solve the initial value problem
y′′=ex/2y′(0)=3y(0)=5y''=e^{x/2} \quad y'(0)=3 \quad y(0)=5
Initial Conditions
Find the function f(x)f(x) whose derivative is 4x−1/x24x-1/x^2 forx>0 x>0 and for which f(1)=−1f(1)=-1.
Solve the initial value problem
f(x)=4x+x22+5,F(1)=1/3\displaystyle f(x)=4x+\frac{x^2}{2}+5, F(1)=1/3
Solve the initial value problem
f(x)=x+x4x2,F(0)=2\displaystyle f(x)=\frac{\sqrt{x}+x^4}{x^2}, F(0)=2
Solve the initial value problem y′=sec⁡2x−e−x,    y(0)=2y'=\sec^2x-e^{-x},\ \ \ \ y\left(0\right)=2
Consider Newton's law of cooling dTdt=2−T10\displaystyle\frac{\text{d}T}{\text{d}t}=2-\frac{T}{10} . Determine T(t)T(t) if T(0)=100T(0)=100
Indefinite Integral
Find f(x)f\left(x\right), given that f′(x)=x2−2+sin⁡xf'\left(x\right)=x^2-2+\sin x and f(0)=3f\left(0\right)=3.
Practice: Rearrange First
Q:\bf{Q:} Solve the initial value problem t2dydt+1=t3t^2\dfrac{dy}{dt} + 1 = t^3 with y(1)=1y(1) = 1
Q:\bf{Q:} Solve the initial value problem y′=x3+sin⁡2xy'=x^3+\sin2x given y(π2)=0y\left(\dfrac{\pi}{2}\right)=0
Q:\bf{Q:}Solve the initial value problem y′=sec⁡2x−e−xy'=\sec^2x-e^{-x} with y(0)=2y\left(0\right)=2
Solve the initial-value problem
y′=x2+1xy(1)=3\begin{aligned} y' &=\dfrac{x^2 + 1}{\sqrt{x}}\\ y(1) & = 3 \end{aligned}
Practice: From Second Derivative to Function
Q:\bf{Q:} Solve the initial-value problem y′′=sin⁡2xy'' = \sin 2x, y′(0)=3y'(0) = 3 , y(0)=5y(0) = 5.
Exponential Functions: $f$ from $f''$
Given f′′(x)=5ex+2sin⁡xf''(x)=5e^x+2\sin x and given f(0)=0f(0)=0, f(π)=0f(\pi)=0, find f(x)f(x). Enter "f(x)=..."
Solve the initial value problem
y′′=ex/2y′(0)=3y(0)=5\begin{aligned} y''& = e^{x/2}\\ y'(0)& = 3\\ y(0) &= 5 \end{aligned}
Solve the initial value problem y′=sec⁡2x−e−x,  y(0)=2y'=\sec^2x-e^{-x},\ \ y\left(0\right)=2.
Find the antiderivative FF
f(x)=x+x4x2,F(1)=2\displaystyle f(x)=\frac{\sqrt{x}+x^4}{x^2}, F(1)=2
Find the antiderivative FF
f(x)=4x+x22+5,F(1)=1/3\displaystyle f(x)=4x+\frac{x^2}{2}+5, F(1)=1/3