Exam-Like Practice Questions

For the following function, find the most general antiderivative.

f(x)=x5+2x4x\displaystyle f(x)=\frac{x^5+2\sqrt{x}}{4x}f(x)=4xx5+2x​​

x510+x1/2+C\dfrac{x^5}{10}+x^{1/2}+C10x5​+x1/2+C

x520−x1/2+C\dfrac{x^5}{20}-x^{1/2}+C20x5​−x1/2+C

x520+x1/2+C\dfrac{x^5}{20}+x^{1/2}+C20x5​+x1/2+C

x510−x1/2+C\dfrac{x^5}{10}-x^{1/2}+C10x5​−x1/2+C

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Find ∫(2x−34−x2+2sin⁡x)dx\displaystyle\int_{ }^{ }\left(2\sqrt{x}-\frac{3}{\sqrt{4-x^2}}+2\sin x\right)dx∫​(2x​−4−x2​3​+2sinx)dx 

Use upper case

C

for any constants

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Compute:  ∫25(x3+sin⁡x)dx\displaystyle \int_{2}^{5}(x^3+\sin{x})\text{d}x∫25​(x3+sinx)dx

Answer

I don't know

Evaluate ∫eex+xdx\int_{ }^{ }e^{e^x+x}dx∫​eex+xdx.

∫e1xx2dx\displaystyle \int_{ }^{ }\frac{e^{\frac{1}{x}}}{x^2}dx∫​x2ex1​​dx

e1x+Ce^{\frac{1}{x}}+Cex1​+C

−e1x+C-e^{\frac{1}{x}}+C−ex1​+C

−e1xx2+C-\frac{e^{\frac{1}{x}}}{x^2}+C−x2ex1​​+C

−3e1xx3+C-\frac{3e^{\frac{1}{x}}}{x^3}+C−x33ex1​​+C

3e1xx3+C\frac{3e^{\frac{1}{x}}}{x^3}+Cx33ex1​​+C

I don't know

Evaluate ∫02(x−2)e(x2−4x)dx\int_0^2\left(x-2\right)e^{\left(x^2-4x\right)}dx∫02​(x−2)e(x2−4x)dx.

Find ∫cot⁡x ln⁡(sin⁡x)dx\int_{ }^{ }\cot x\ \ln\left(\sin x\right)dx∫​cotx ln(sinx)dx.

[ln⁡(sin⁡x)]22+C\frac{\left[\ln\left(\sin x\right)\right]^2}{2}+C2[ln(sinx)]2​+C

csc⁡x[ln⁡(sin⁡x)]22+C\frac{\csc x\left[\ln\left(\sin x\right)\right]^2}{2}+C2cscx[ln(sinx)]2​+C

sin⁡x[ln⁡x]22+C\frac{\sin x\left[\ln x\right]^2}{2}+C2sinx[lnx]2​+C

sec⁡x ln⁡(sin⁡x)+C\sec x\ \ln\left(\sin x\right)+Csecx ln(sinx)+C

None of the above

I don't know

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Compute
∫142x2  dx\int_{1}^{4} \frac{2}{x^2} \; dx∫14​x22​dx

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Estimate 
∫12(x+1)dx\int_1^2\left(x+1\right)dx∫12​(x+1)dx
using left Riemann sum with 10 subintervals.

Answer

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Calculate the right endpoint and left endpoint Riemann sums for the function f(x)=x2−1f\left(x\right)=x^2-1f(x)=x2−1  over the interval  [0, 3]\left[0,\ 3\right][0, 3] by dividing the interval into 6 subintervals of equal length (i.e. use 6 rectangles)

Lower Sum

Upper Sum

I don't know

🌶️ Evaluate the limit by recognizing the Riemann sum:
lim⁡n→∞∑i=1n27i3n4.\lim\limits_{n\rightarrow\infin}\displaystyle\sum_{i=1}^n\frac{27i^3}{n^4}.n→∞lim​i=1∑n​n427i3​.
Enter your answer as a fraction.

Answer

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Rewrite lim⁡n→∞∑i=1n12isin⁡(2in)\displaystyle \lim_{n\to\infty}\sum_{i=1}^n\frac{1}{2i}\sin\left(\frac{2i}{n}\right)n→∞lim​i=1∑n​2i1​sin(n2i​) as a definite integral

∫02xsin⁡x dx\displaystyle \int_0^2x\sin x\ dx∫02​xsinx dx

12∫02xsin⁡x dx\displaystyle \frac{1}{2}\int_0^2x \sin x\ dx21​∫02​xsinx dx

∫02sin⁡x2xdx\displaystyle \int_0^2\frac{\sin x}{2x}dx∫02​2xsinx​dx

2∫02sin⁡xxdx2\displaystyle \int_0^2\frac{\sin x}{x}dx2∫02​xsinx​dx

4∫02sin⁡xxdx4\displaystyle \int_0^2\frac{\sin x}{x}dx4∫02​xsinx​dx

I don't know

Evaluate lim⁡n→∞∑i=1n (3n)11+3in\displaystyle\lim_{n\rightarrow\infty}\sum_{i=1}^n\ \left(\frac{3}{n}\right)\frac{1}{\sqrt{1+\dfrac{3i}{n}}}n→∞lim​i=1∑n​ (n3​)1+n3i​​1​

Answer

I don't know

Practice: Definite Integral
Evaluate ∫0319+x2+2xln⁡2 dx\int_0^3\frac{1}{9+x^2}+2^x\ln2\ dx∫03​9+x21​+2xln2 dx.

Evaluate ∫02(x−2)e(x2−4x)dx\int_0^2\left(x-2\right)e^{\left(x^2-4x\right)}dx∫02​(x−2)e(x2−4x)dx.

Integrate
∫−π/2π/2sin⁡3(x)cos⁡(x)  dx\int_{- \pi/2}^{\pi/2} \sin^{3}(x) \cos(x) \; dx∫−π/2π/2​sin3(x)cos(x)dx

Answer

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Compute ∫081x3dx.\int_0^8\frac{1}{\sqrt[3]{x}}dx.∫08​3x​1​dx.

Answer

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Let f(x)=∫02x1+4t  dt.f\left(x\right)=\int_0^{2x}\sqrt{1+4t} \; dt.f(x)=∫02x​1+4t​dt. Compute f′(2).f'(2).f′(2). 

Answer

I don't know

If g(x)=∫1xcos⁡t dtg\left(x\right)=\int_1^x\cos t\ dtg(x)=∫1x​cost dt, find g′(π2)g'\left(\frac{\pi}{2}\right)g′(2π​).

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Find ddx(∫xx2t2dt)\frac{d}{dx}\left(\int_x^{x^2}t^2dt\right)dxd​(∫xx2​t2dt).

I have answered this question

Wize University Calculus 1 Textbook > Integrals

Popular Courses

$\displaystyle \int_{ }^{ }\frac{e^{\frac{1}{x}}}{x^2}dx$

Practice: Definite Integral

Wize University Calculus 1 Textbook > Integrals

Exam-Like Practice Questions

Popular Courses

∫e1xx2dx\displaystyle \int_{ }^{ }\frac{e^{\frac{1}{x}}}{x^2}dx∫​x2ex1​​dx

Practice: Definite Integral

$\displaystyle \int_{ }^{ }\frac{e^{\frac{1}{x}}}{x^2}dx$