Question

[tex] \int\limits^4_1 (x²+1/x)⁻¹=.....,Select correct option from the given options.(a) log(17/2)(b) 1/2 log(17/2)(c) 2 log(17)(d) log(17) [\tex]

Answer 1

Answer:

The correct option is (c).

i.e;

$\dfrac{1}{2}log(\dfrac{17}{2})$.

Step-by-step explanation:

Given equation is:

$\displaystyle\int _1^4(\dfrac{x^2+1}{x})^{-1}dx$

We may write the equation as:

$\displaystyle\int _1^4(\dfrac{x}{x^2+1})dx$

Now substituting:

$x^2+1=z$

$2x\,dx=dz\\x\,dx=\dfrac{dz}{2}$

Substituting for the value of limits:

Lower limit:

When $x=1$

$z=1+1\\z=2$

Upper limits:

When $x=4$

$z=(4)^2+1\\z=17$

Hence substituting the values of limits:

$=\displaystyle\int _2^{17}\dfrac{1}{z}\cdot \dfrac{dz}{2}$

$=\dfrac{1}{2}[log(z)]_2^{17}$

$=\dfrac{1}{2}[log(17)-log(2)]$

$=\dfrac{1}{2}log(\dfrac{17}{2})$