 # Master the Art of Integration by Parts: Comprehensive Examples and Expert Advice

Recent questions in Integration by Parts Shirley Peck 2023-03-23

## Integration of 1/sinx-sin2x dx bedastega4n3 2023-03-13

## Use the summation formulas to rewrite the expression $\Sigma \frac{2i+1}{{n}^{2}}$ as i=1 to n without the summation notation and then use the result to find the sum for n=10, 100, 1000, and 10000. piteraufqvw 2023-02-15

## Find the sum of the infinite geometric series $\Sigma {\left(\frac{1}{10}\right)}^{n}$ from n=0 to $\infty$ Teresa Manning 2023-02-04

## Define $f\left(x\right)=$ integral from $0\mathrm{to}x\mathrm{sin}\left(t\right)dt,x\ge 0$. ThenA) F is increasing only in the interval $\left(0,\frac{\pi }{2}\right)$ B) f is decreasing in the interval $\left(0,\pi \right)$ C) f attains maximum at $x=\frac{\pi }{2}$ D) f attains minimum at $x=\pi$E) f attains maximum at $x=\pi$ Justine Pennington 2022-11-29

## Integration of ${x}^{n}{e}^{-x}dx$The question is a definite integral${\int }_{0}^{\mathrm{\infty }}\frac{{x}^{n}}{{e}^{x}}\phantom{\rule{thinmathspace}{0ex}}dx$So, I'm integrating it by parts, and going by the LAITE principle, I get:${I}_{n}=\frac{-{x}^{n}}{{e}^{x}}+{\int }_{0}^{\mathrm{\infty }}\frac{n{x}^{n-1}}{{e}^{x}}\phantom{\rule{thinmathspace}{0ex}}dx$The value inside the integral is$n{I}_{n-1}$, so all I'm left is to get the limit of the first term as x ranges from 0 to infinity.$\sum \frac{{x}^{n}}{{e}^{x}}$Somehow, I'm not exactly sure why I feel like the limit of the first term would go to zero (or n!), given that formula, but I'm not exactly sure how to substitute the n! formula for ${e}^{x}$. How do I proceed from here?The options are$n!-n{I}_{n-1}$$n!+n{I}_{n-1}$$n{I}_{n-1}$none of these Jack Ingram 2022-10-29

## Find the area of the surface generated by revolving the curve $x=\frac{{y}^{3}}{3},0⇐y⇐5$, about the y axis. The area of the surface generated by revolving the curve $x=\frac{{y}^{3}}{3},0⇐y⇐5$, about the y-axis is __ square units okeleyeoyindamola2006 2022-08-05

##  klepkowy7c 2022-07-28

## Use integration by parts to prove the reduction formula.$\int {x}^{n}{e}^{x}dx={x}^{n}{e}^{x}-n\int {x}^{n-1}{e}^{x}dx$ Ciara Rose 2022-07-26

## Whats the integral of $x{e}^{-x}$ Spencer Lutz 2022-05-13

## Integrate $\frac{5}{8}﻿\mathrm{ln}\left(\frac{3}{{x}^{2}}\right)$ with respect to x. vilitatelp014 2022-05-13

## Integrate $\frac{5}{4}t{\mathrm{sin}}^{-1}\left(t\right)$ with respect to t. garcialdaria2zky1 2022-05-12

## Compute $\int \frac{6}{5}﻿{t}^{2}\mathrm{ln}\left(\frac{1}{t}\right)dt$. Yasmine Larson 2022-05-12

## Evaluate $\int \frac{8}{7}﻿{x}^{2}\mathrm{ln}\left(x{\right)}^{2}dx$. arbixerwoxottdrp1l 2022-05-12

## Evaluate $\int \frac{9}{7}{\mathrm{tan}}^{-1}\left(x\right)dx$. Matilda Webb 2022-05-12

## Evaluate $\int \frac{3t}{5\sqrt{2t-1}}﻿dt$. agrejas0hxpx 2022-05-12

## Evaluate $\int \frac{6t}{7\sqrt{3t+2}}﻿dt$. Jazlyn Raymond 2022-05-12

## Integrate $\frac{1}{4}﻿{x}^{5}\mathrm{ln}\left(x{\right)}^{2}$ with respect to x. Brody Collins 2022-05-12

## Evaluate $\int \frac{5}{4}{x}^{3}\mathrm{ln}\left(x{\right)}^{3}dx$. kwisangqaquqw3 2022-05-12
## Evaluate $\int \frac{5}{9}{\mathrm{cos}}^{-1}\left(t\right)dt$. Merati4tmjn 2022-05-12