$$ \begin{array}{l} ext { Use a change of variables to evaluate the following definite integral. }\ \int_{0}^{1} \frac{2 x}{\left(x^{2}+5 ight)^{3}} d x \end{array} $$

Question

$$ \begin{array}{l} 	ext { Use a change of variables to evaluate the following definite integral. }\ \int_{0}^{1} \frac{2 x}{\left(x^{2}+5ight)^{3}} d x \end{array} $$

UpStudy AI · Accepted Answer

Let
$$
u=x^2+5
$$
so that
$$
\frac{du}{dx}=2x\quad\Longrightarrow\quad du=2x\,dx.
$$
When $$ x=0 $$, we have:
$$
u=0^2+5=5,
$$
and when $$ x=1 $$, we have:
$$
u=1^2+5=6.
$$

Thus, the integral becomes
$$
\int_{0}^{1}\frac{2x}{\left(x^2+5\right)^3}dx=\int_{5}^{6}\frac{1}{u^3}du=\int_{5}^{6} u^{-3}\,du.
$$

Now, we evaluate the integral:
$$
\int u^{-3}\,du=\frac{u^{-2}}{-2}+C=-\frac{1}{2u^2}+C.
$$

Applying the limits from $$ u=5 $$ to $$ u=6 $$ gives:
$$
\left[-\frac{1}{2u^2}\right]_{5}^{6}=-\frac{1}{2(6)^2}+\frac{1}{2(5)^2}=-\frac{1}{72}+\frac{1}{50}.
$$

To combine the fractions, find a common denominator:
$$
\frac{1}{50}-\frac{1}{72}=\frac{72-50}{50\cdot72}=\frac{22}{3600}=\frac{11}{1800}.
$$

Thus, the value of the integral is
$$
\frac{11}{1800}.
$$
The integral evaluates to $$ \frac{11}{1800} $$.

Solución de inteligencia artificial de Upstudy

Responder

Solución

Mind Expander

preguntas relacionadas

Latest Calculus Questions