# arrow_back Evaluate: $\int_{1}^{3}\left(\frac{x-1}{(x+1)^{2}}\right) \mathrm{d} x$

93 views
Evaluate: $\int_{1}^{3}\left(\frac{x-1}{(x+1)^{2}}\right) \mathrm{d} x$

Explanation
$\int_{1}^{3}\left(\frac{x-1}{(x+1)^{2}}\right) \mathrm{d} x$
Let $x+1=u, x-1=u-2$
When $x=3, u=3+1=4$
\begin{aligned} &x=1, u=1+1=2 \\ &\therefore \int_{1}^{3}\left(\frac{x-1}{(x+1)^{2}}\right) \mathrm{d} x \equiv \int_{2}^{4}\left(\frac{u-2}{u^{2}}\right) \mathrm{d} u \\ &=\int_{2}^{4}\left(\frac{u}{u^{2}}-\frac{2}{u^{2}}\right) \mathrm{d} u \\ &=\int_{2}^{4}\left(\frac{1}{u}-2 u^{-2}\right) \mathrm{d} u \\ &=\left.\left[\ln u-\frac{2 u^{-1}}{-1}\right]\right|_{2} ^{4} \\ &=\left.\left[\ln u+2 u^{-1}\right]\right|_{2} ^{4} \\ &=\left[\ln u+\frac{2}{u}\right]_{2}^{4} \\ &=\left(\ln 4+\frac{2}{4}\right)-\left(\ln 2+\frac{2}{2}\right) \\ &=\ln 4-\ln 2-\frac{1}{2} \\ &=\ln \left(\frac{4}{2}\right)-\frac{1}{2} \\ &=\ln 2-\frac{1}{2} \end{aligned}
by Platinum
(119,140 points)

## Related questions

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
Evaluate $\int_{-1}^{0}(x+1)(x-2) \mathrm{d} x$
Evaluate $\int_{-1}^{0}(x+1)(x-2) \mathrm{d} x$Evaluate $\int_{-1}^{0}(x+1)(x-2) \mathrm{d} x$ A. $\frac{7}{6}$ B. $\frac{5}{6}$ C. $\frac{-5}{6}$ D. $\frac{-7}{6}$ ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
Evaluate the integral $\int_{0}^{1} \left ( 4x-1 \right )^{3}dx$
Evaluate the integral $\int_{0}^{1} \left ( 4x-1 \right )^{3}dx$Evaluate the integral $\int_{0}^{1} \left ( 4x-1 \right )^{3}dx$ ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
$\int\left(\frac{1}{x^{2}}+\frac{6}{x^{3}}\right) d x$
$\int\left(\frac{1}{x^{2}}+\frac{6}{x^{3}}\right) d x$$\int\left(\frac{1}{x^{2}}+\frac{6}{x^{3}}\right) d x ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Estimate \int_{0}^{\sqrt{\pi}} \sin \left(x^{2}\right) d x using Gaussian quadrature. 0 answers 152 views Estimate \int_{0}^{\sqrt{\pi}} \sin \left(x^{2}\right) d x using Gaussian quadrature.Estimate \int_{0}^{\sqrt{\pi}} \sin \left(x^{2}\right) d x using Gaussian quadrature. ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Evaluate \int_{0}^{1}(4 x-1)^{3} d x 1 answer 64 views Evaluate \int_{0}^{1}(4 x-1)^{3} d xEvaluate \int_{0}^{1}(4 x-1)^{3} d x ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Evaluate the integral \int_{0}^{1} x^{2} d x 1 answer 93 views Evaluate the integral \int_{0}^{1} x^{2} d xEvaluate the integral \int_{0}^{1} x^{2} d x ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 \int \frac{1}{2 \mathrm{x}} d x 1 answer 106 views \int \frac{1}{2 \mathrm{x}} d x$$\int \frac{1}{2 \mathrm{x}} d x$ ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
$\int\left(\frac{4}{x}-e^{x}\right) d x$
$\int\left(\frac{4}{x}-e^{x}\right) d x$$\int\left(\frac{4}{x}-e^{x}\right) d x ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 \int\left(x^{-1}-1\right) d x 1 answer 110 views \int\left(x^{-1}-1\right) d x$$\int\left(x^{-1}-1\right) d x$ ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
Evaluate $\int_{0}^{1} \dfrac{\ln x}{\sqrt{1+x^{4}}}$
Evaluate $\int_{0}^{1} \dfrac{\ln x}{\sqrt{1+x^{4}}}$Evaluate $\int_{0}^{1} \dfrac{\ln x}{\sqrt{1+x^{4}}}$ ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
$\int\left(5 x^{2}-7 x+6\right) d x$
$\int\left(5 x^{2}-7 x+6\right) d x$$\int\left(5 x^{2}-7 x+6\right) d x ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 \int\left(x^{2}-2 x\right) d x 1 answer 120 views \int\left(x^{2}-2 x\right) d x$$\int\left(x^{2}-2 x\right) d x$ ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
Show that $\frac{d}{d x}\left(\sin ^{-1} x\right)=\frac{1}{\sqrt{1-x^{2}}}$
Show that $\frac{d}{d x}\left(\sin ^{-1} x\right)=\frac{1}{\sqrt{1-x^{2}}}$Show that $\frac{d}{d x}\left(\sin ^{-1} x\right)=\frac{1}{\sqrt{1-x^{2}}}$ ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
Integrate $\int x^{\frac{1}{3}} d x$
Integrate $\int x^{\frac{1}{3}} d x$Integrate $\int x^{\frac{1}{3}} d x$ ...
The number of $x \in[0,2 \pi]$ for which $\left|\sqrt{2 \sin ^{4} x+18 \cos ^{2} x}-\sqrt{2 \cos ^{4} x+18 \sin ^{2} x}\right|=1$ is:
The number of $x \in[0,2 \pi]$ for which $\left|\sqrt{2 \sin ^{4} x+18 \cos ^{2} x}-\sqrt{2 \cos ^{4} x+18 \sin ^{2} x}\right|=1$ is:The number of $x \in0,2 \pi$ for which $\left|\sqrt{2 \sin ^{4} x+18 \cos ^{2} x}-\sqrt{2 \cos ^{4} x+18 \sin ^{2} x}\right|=1$ is: ...