Recent questions in Integration by Parts

Integral CalculusAnswered question

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.

Integral CalculusAnswered question

piteraufqvw 2023-02-15

Find the sum of the infinite geometric series $\Sigma {\left(\frac{1}{10}\right)}^{n}$ from n=0 to $\infty$

Integral CalculusAnswered question

Teresa Manning 2023-02-04

Define $f\left(x\right)=$ integral from $0\mathrm{to}x\mathrm{sin}\left(t\right)dt,x\ge 0$. Then

A) 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 $

A) 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 $

Integral CalculusAnswered question

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

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

Integral CalculusAnswered question

Jack Ingram 2022-10-29

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

Integral CalculusAnswered question

klepkowy7c 2022-07-28

Use integration by parts to prove the reduction formula.

$\int {x}^{n}{e}^{x}dx={x}^{n}{e}^{x}-n\int {x}^{n-1}{e}^{x}dx$

$\int {x}^{n}{e}^{x}dx={x}^{n}{e}^{x}-n\int {x}^{n-1}{e}^{x}dx$

Integral CalculusAnswered question

Spencer Lutz 2022-05-13

Integrate $\frac{5}{8}\ufeff\mathrm{ln}(\frac{3}{{x}^{2}})$ with respect to x.

Integral CalculusAnswered question

vilitatelp014 2022-05-13

Integrate $\frac{5}{4}t{\mathrm{sin}}^{-1}(t)$ with respect to t.

Integral CalculusAnswered question

garcialdaria2zky1 2022-05-12

Compute $\int \frac{6}{5}\ufeff{t}^{2}\mathrm{ln}(\frac{1}{t})dt$.

Integral CalculusAnswered question

Yasmine Larson 2022-05-12

Evaluate $\int \frac{8}{7}\ufeff{x}^{2}\mathrm{ln}(x{)}^{2}dx$.

Integral CalculusAnswered question

arbixerwoxottdrp1l 2022-05-12

Evaluate $\int \frac{9}{7}{\mathrm{tan}}^{-1}(x)dx$.

Integral CalculusAnswered question

Matilda Webb 2022-05-12

Evaluate $\int \frac{3t}{5\sqrt{2t-1}}\ufeffdt$.

Integral CalculusAnswered question

agrejas0hxpx 2022-05-12

Evaluate $\int \frac{6t}{7\sqrt[3]{3t+2}}\ufeffdt$.

Integral CalculusAnswered question

Jazlyn Raymond 2022-05-12

Integrate $\frac{1}{4}\ufeff{x}^{5}\mathrm{ln}(x{)}^{2}$ with respect to x.

Integral CalculusAnswered question

Brody Collins 2022-05-12

Evaluate $\int \frac{5}{4}{x}^{3}\mathrm{ln}(x{)}^{3}dx$.

Integral CalculusAnswered question

kwisangqaquqw3 2022-05-12

Evaluate $\int \frac{5}{9}{\mathrm{cos}}^{-1}(t)dt$.

Integral CalculusAnswered question

Merati4tmjn 2022-05-12

Find the integral: $\int \frac{3}{2}\ufeff{t}^{2}{\mathrm{sin}}^{-1}(t)dt$.

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An interesting concept is based on a special method where you have to use two functions by multiplying them together. When you are dealing with equations you should focus on differentiation, integration, and various simplification methods. Speaking of definite integration by parts, you will be able to find the answers and seek similar questions based on your objectives. There is also integration by parts examples that are based on the word problems and the formulas. The best solution is to reverse-engineer the equations based on the solutions. Take your time to explore and the integral calculus challenges will become easier.