Evaluate the definite integral in Exercise: _ 2 ^ 3 x dx x^ 2 +1 - Vidyadip

Question

Evaluate the definite integral in Exercise:
$\int\limits_{2}^{3}\frac{\text{x}\ \text{dx}}{\text{x}^{2}+1}$

✓

Answer

$\text{Let}\text{I}=\int\limits_{2}^{3}\frac{\text{x}}{\text{x}^{2}+1}\text{dx}$
$\int\frac{\text{x}}{\text{x}^{2}+1}\text{dx}=\frac{1}{2}\int\frac{2\text{x}}{\text{x}^{2}+1}\text{dx}=\frac{1}{2}\text{log}(1+\text{x}^{2})=\text{F}\text{(x)}$
By second fundamental theorem of calculus, we obtain
$\text{I}=\text{F}(3)-\text{F}(2)$
$=\frac{1}{2}\big[\text{log}\big(1+(3)^{2}\big)-\text{log}\big(1+(2)^{2}\big)\big]$
$=\frac{1}{2}\big[\text{log}(10)-\text{log(5)}\big]$
$=\frac{1}{2}\text{log}\bigg(\frac{10}{5}\bigg)=\frac{1}{2}\text{log}2$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "2 Marks Questions" from INTEGRALS→INTEGRALS — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Write the order of the differential equation $\text{y}=\text{x}\frac{\text{dy}}{\text{dx}}+\text{a}\sqrt{1+\Big(\frac{\text{dy}}{\text{dx}}\Big)^{2}}.$

If $\frac{\pi}{2}\leq\text{x}\leq\frac{3\pi}{2}$ and $\text{y}=\sin^{-1}(\sin\text{x}),$ find $\frac{\text{dy}}{\text{dx}}.$

Evaluate the following:
$\sin^{-1}\Big(\sin\frac{17\pi}{8}\Big)$

What is the principal value of $\sin^{-1}\Big(-\frac{1}{2}\Big)?$

If $\text{f(x)}=\begin{cases}\frac{\text{x}^2-16}{\text{x}-4},&\text{if }\text{ x}\neq4\\\text{k},&\text{if }\text{ x}=4\end{cases}$ is continuous at x = 4, find k.

A and B are two events such that P (A) $\ne$ 0. Find P(B|A), if A is a subset of B.

Find that point on curve $y=x^2-2 x+3$, the tangent drawn on that is parallel to line $2 \mathrm{x}-\mathrm{y}+9=0$.

If $\vec{\text{a}},\vec{\text{b}}$ are two vectors, then write the truth value of the following statement:$\big|\vec{\text{a}}\big|=\big|\vec{\text{b}}\big|\Rightarrow\vec{\text{a}}=\vec{\text{b}}$

If $\vec{\text{r}}=\text{x}\hat{\text{i}}+\text{y}\hat{\text{j}}+\text{z}\hat{\text{k}},$ then write the value of $\big|\vec{\text{r}}\times\hat{\text{i}}\big|^2.$

Let $R_0$ denote the set of all non$-$zero real numbers and let $A = R_0 \times R_0.$ If$ '*' $is a binary operation on adefined by, $(a, b) * (c, d) = (ac, bd)$ for all $(a, b), (c, d) \in A $ Find the invertible element in $A.$