# Thread:

1. ## Using integral by parts

Hello everyone it's me again...

For this question I have to use integral by parts to find the integral of (t+5)/(sqrt 8-t)

I have:
u= t+5
v' = (8-t)^-1/2

u'= 1
v= 2(8-t)^1/2

then I plugged in the numbers

(t+5)(2(8-t))^1/2 = integral (8-t)^1/2

I don't really know what to do could someone help me out? /:

2. ## Re: Using integral by parts

Integration by Parts is not the best method to use here.

$\displaystyle \begin{align*} \int{\frac{t+5}{\sqrt{8-t}}\,dt} = -\int{-\frac{t+5}{\sqrt{8-t}}\,dt} \end{align*}$

Let $\displaystyle \begin{align*} u = 8 - t \implies du = -dt \end{align*}$ and note that $\displaystyle \begin{align*} t + 5 = 13 - u \end{align*}$ and the integral becomes

$\displaystyle \begin{align*} -\int{-\frac{t + 5}{\sqrt{8-t}}\,dt} &= -\int{\frac{13 - u}{\sqrt{u}}\,du} \\ &= -\int{13u^{-\frac{1}{2}} - u^{\frac{1}{2}} \, du} \end{align*}$

This should now be easy to solve.

3. ## Re: Using integral by parts

Originally Posted by Prove It
Integration by Parts is not the best method to use here.

$\displaystyle \begin{align*} \int{\frac{t+5}{\sqrt{8-t}}\,dt} = -\int{-\frac{t+5}{\sqrt{8-t}}\,dt} \end{align*}$

Let $\displaystyle \begin{align*} u = 8 - t \implies du = -dt \end{align*}$ and note that $\displaystyle \begin{align*} t + 5 = 13 - u \end{align*}$ and the integral becomes

$\displaystyle \begin{align*} -\int{-\frac{t + 5}{\sqrt{8-t}}\,dt} &= -\int{\frac{13 - u}{\sqrt{u}}\,du} \\ &= -\int{13u^{-\frac{1}{2}} - u^{\frac{1}{2}} \, du} \end{align*}$

This should now be easy to solve.
Unfortunately I have to use integration by parts as instructed by my teacher but this should help me come up with the answer. Thank you.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 8, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9553149342536926, "perplexity_flag": "middle"}