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What is the integral of (5(2x))^4 dx?
Indefinite Integral required
What is the integral of 3e^(27x) dx?
Indefinite Integral required
What is the integral of 5 sinh(3x + 1) dx?
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What is the integral of sin(5t^2) dt?
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What is the integral of (14x^5 - 10x^3 - 2) dx?
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What is the integral of x/(e^x) dx?
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Given the values: ∫ from 0 to 20 of f(x)dx = 7, ∫ from 0 to 20 of f(x)dx = -3, and ∫ from 0 to 20 of g(x)dx = 5. What is the evaluation of ∫ from 13 to 20 of f(x)dx?
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What is the average value of f(x) = e^(3x) on the interval [1, 4]?
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What is the r.m.s value of i(t) = 2 cos(4t) across the interval [0, π/2]?
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Study Notes
Assignment Overview
- Assignment 7 is for the course ENGR121.
- Due date is 11:59 PM on Wednesday, May 22, 2024.
- Submission must be a single PDF file named with the user's username.
Submission Guidelines
- Use exact forms for numbers; avoid decimal approximations.
- For example, use symbols like π, e, and specific integers rather than their numerical representations.
- Review submissions after upload to confirm accuracy.
Indefinite Integrals
- Tasks include finding indefinite integrals for functions such as:
- (a) ( \int (5(2x)^4) , dx )
- (b) ( \int 3x^{4/16} , dx )
- (c) ( \int 3e^{27x} , dx )
- Other integrals include:
- (d) ( \int (14x^5 - 10x^3 - 2) , dx )
- (e) ( \int 5 \sinh(3x + 1) , dx )
- (f) ( \int \sin(5t^2) , dt )
Definite Integrals with Given Values
- Given values include ( \int_0^7 f(x) , dx = 7 ), ( \int_0^{20} f(x) , dx = -3 ), and ( \int_0^{20} g(x) , dx = 5 ).
- Evaluate:
- (a) ( \int_{13}^{20} f(x) , dx )
- (b) ( \int_0^{20} 7g(x) , dx )
- (c) ( \int_0^{20} (2g(x) - 3f(x)) , dx )
- (d) ( \int_{13}^{20} 4f(x) , dx )
Additional Definite Integrals
- Tasks include evaluating:
- (a) ( \int_0^7 \ln(6) , dx )
- (b) ( \int_0^{\pi} \sin(6x + 3) , dx )
- (c) ( \int_0^4 6(3t)^4 , dt )
- Also includes more complex integrals:
- (d) ( -2 \int_0^{\pi} (4e^{-7x} + e^4) , dx )
- (e) ( \int_0^1 (4x^4 - 3x^3 + 5x - 7) , dx )
- (f) ( -\pi \int_0^{\pi} |\sin(x)| , dx )
Improper Integrals
- Tasks involve evaluating improper integrals:
- (a) ( \int_{-\infty}^0 x e^x , dx )
- (b) ( \int_{-10}^{10} \frac{1}{x + 3} , dx )
Integration by Substitution
- Perform substitution to solve:
- (a) ( \int_0^1 5x^{2/3}(x + 7) , dx )
- (b) ( \int \cos(\ln(7x) + 3) , dx )
- Further tasks include integration of:
- (c) ( \int e^{-4 \sin(x)} , dx )
- (d) ( \int \frac{3}{2x^4 + 3} , dx )
- (e) ( \int x(4x + 7)^6 , dx )
Average and RMS Values
- Calculate the average value of ( f(x) = e^{3x} ) over the interval [1, 4].
- Determine the root mean square (RMS) value of ( i(t) = 2 \cos(4t) ) across the interval [0, ( \frac{\pi}{2} )].
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Description
This document outlines the requirements for ENGR121 Assignment 7 due on May 22, 2024. It emphasizes the need for submissions to be in a single PDF format and to maintain exact numeric forms for mathematical constants. Follow the guidelines to ensure a successful submission.
