Lectures>Practice

# Exercises

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## Exercises

Evaluate the following definite integrals.

 $\int_1^3 x dx$ $\int_{-2}^2 x dx$ $\int_1^3 \frac{x}{2} dx$ $\int_1^3 \frac{x}{4} dx$ $\int_0^4 (x + 1) dx$ $\int_{-1}^2 (x^2 + x - 1) dx$ $\int_{1}^4 (x^3 - 1) dx$ $\int_{-1}^1 x^{2/3} dx$ $\int_{1}^3 x^{-4/5} dx$ $\int_{-1}^1 \frac{(x+1)(x-2)}{x^2} dx$ $\int_1^4 \frac{1}{x} dx$ $\int_1^e \frac{1}{2x} dx$ $\int_1^{e^2} \frac{x+1}{x} dx$ $\int_2^5 \frac{1}{x-1} dx$ $\int_2^4 \frac{x+1}{x-1} dx$ $\int_{1}^3 (\frac{3}{x} + \frac{x}{3}) dx$ $\int_{1}^4 (x + \sqrt{x}) dx$ $\int_{1}^9 \frac{x + \sqrt{x}}{\sqrt{x}} dx$ $\int_{1}^4 (\frac{3}{x} + \frac{x}{3}) dx$ $\int_0^1 e^x dx$ $\int_0^2 e^{x+1} dx$ $\int_0^3 3e^{2x} dx$ $\int_{-4}^1 2^{-2x} dx$ $\int_{-3}^3 3^{2x-1} dx$ $\int_0^{\pi} \sin x dx$ $\int_{-\pi}^{\pi} \sin x dx$ $\int_{-\pi}^{\pi} \cos x dx$ $\int_{-\pi/2}^{\pi/2} (\cos x + \sin x) dx$ $\int_{0}^{\pi/2} 2\sin (3x) dx$ $\int_{-1}^{2} |x| dx$ $\int_{0}^{3} |x - 1| dx$ $\int_{-1}^{2} \cosh x dx$ $\int_{0}^{3} \sinh (3x) dx$

Evaluate the following expressions.

 $\frac{d}{dx}\int_{0}^x (3x) dx$ $\frac{d}{dx}\int_{-2}^x e^{-t} dt$ $\frac{d}{dt}\int_{-2}^t \ln(u) du$ $\frac{d}{dx}\int_{2}^{x^2} (x + 1) dx$ $\frac{d}{dx}\int_{5}^{\sqrt{x}} (x^2 + x - 1) dx$ $\frac{d}{dx}\int_{-2}^{e^x} \frac{1}{x} dx$

Find the average value of the following functions on the given interval.

 $f(x) = \cos x$ on $[0, \pi]$ $f(x) = 2\cos x$ on $[0, \pi]$ $f(x) = \cos 2x$ on $[0, \pi]$ $f(x) = e^{-x}$ on $[-1, 1]$ $f(x) = \frac{1}{x+1}$ on $[0, 3]$ $f(x) = x^2 + 1$ on $[-2, 2]$

## Explorations

The following questions explore the affect of various transformations of a function on the function's definite integral.

1. Dilations If $f$ is continuous and $\int_0^{10} f(x)dx = 12$ then what is $\int_0^5 f(2x) dx$?
2. Dilations If $f$ is continuous and $\int_0^{a} f(x)dx = b$ then what is $\int_0^{a/n} f(nx) dx$?
3. Reflections If $f$ is continuous, show that $\int_a^{b} f(x) = \int_{-b}^{-a} f(-x) dx$ by comparing areas.
4. Translations If $f$ is continuous, show that $\int_a^{b} f(x + c) = \int_{a+c}^{b+c} f(x) dx$ by comparing areas.
5. Translations If $f$ is continuous, show that $\int_a^{b} (f(x) + c) = \int_{a}^{b} f(x) dx + c(b - a)$.

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