ENGR121_Assignment_7.pdf

ENGR121 Assignment 7 DUE: 11:59pm Wednesday 22 May 2024 Submission is online via the Submission link in the web left bar. Ensure your submission is a single pdf file, with a name that ends with the characters yourUserName.pdf. View your...

ENGR121 Assignment 7 DUE: 11:59pm Wednesday 22 May 2024 Submission is online via the Submission link in the web left bar. Ensure your submission is a single pdf file, with a name that ends with the characters yourUserName.pdf. View your submission after uploading it. Please keep√every number in your work in its exact form. For examples, please write π instead of 3.1415..., 2 instead of 1.4142..., 34 instead of 1.333..., and even 21 instead of 0.5, etc. Please also use 43 instead of 1 13 , as the later bounds to be confused with 1 · 13. 1. Find the following Indefinite Integrals: R √ R R (a) ( 5 2x)4 dx (c) 3e27x dx (e) 5 sinh(3x + 1)dx (b) 3x416 dx sin 5t2 dt R R R (d) (14x5 − 10x3 − 2)dx (f) R 13 R 20 R 20 2. Suppose 0 f (x)dx = 7, 0 f (x)dx = −3 and 0 g(x)dx = 5. Evaluate the following Definite Integrals (Show All Working): R 20 R 20 (a) 13 f (x)dx (c) 0 (2g(x) − 3f (x))dx R 20 R0 (b) 0 7g(x)dx (d) 13 4f (x)dx 3. Evaluate the following Definite Integrals (Show All Working): R7 R4 √ R1 (a) 0 ln(6)dx (c) 6 ( 3 t)4 dt (e) 0 (4x4 − 3x3 + 5x − 7)dx Rπ R0 Rπ (b) 0 sin(6x + 3)dx (d) −2 (4e−7x + e4 )dx (f) −π | sin(x)|dx 4. Find the following improper integrals. Show all working Z 0 Z 10 x 1 (a) e dx (b) 3 dx −∞ −10 (x + 3) 5. Use substitution to solve the following Integrals. Show All Working. Do not simplify your answer. R 1 5x2 R −4 sin(x) R 5 6 (a) 0 √ 5 3 x +7 dx (c) 3 cos(x)e dx (e) x (4x + 7)6 dx R5 R x3 (b) cos(ln(7x)+3) dx (d) dx 4 x 2x4 + 3 6. What is the average value f (x) = e3x on the interval [1, 4]? Show All Working. 7. Find the r.m.s value of i(t) = 2 cos(4t) across [0, π2 ]? Show your working. 1