Evaluate the following definite integrals: _ 1 ^2 dx - Vidyadip

Question

Evaluate the following definite integrals:
$\int_{1}^\limits{2}\log\text{x}\text{ dx}$

✓

Answer

Let $\text{I}=\int_{1}^\limits{2}\log\text{x}\text{ dx}$ Then,
$\text{I}=\int_{1}^\limits{2}1\log\text{x}\text{ dx}$
Integrating by parts.
$\Rightarrow\text{I}=\big[\text{x }\log\text{ x}\big]^2_1-\int_{1}^\limits{2}\frac{1}{\text{x}}\text{x}\text{ dx}$
$\Rightarrow\text{I}=\big[\text{x }\log\text{ x}\big]^2_1-\int_{1}^\limits{2}\text{dx}$
$\Rightarrow\text{I}=\big[\text{x }\log\text{ x}\big]^2_1-\big[\text{x}\big]^2_1$
$\Rightarrow\text{I}=2\log2-2+1$
$\Rightarrow\text{I}=2\log2-1$

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 "3 Marks Question" from Definite Integrals→Definite 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 a value of $\int\frac{\text{a}^{\text{x}}}{3+\text{a}^{\text{x}}}\text{ dx}$

Show that $\text{AB}\neq\text{BA}$ in the following cases:
$\text{A}=\begin{bmatrix}5&-1\\6&7\end{bmatrix}$ and $\text{B}=\begin{bmatrix}2&1\\3&4\end{bmatrix}$

Integrate the function $\frac{3 x}{1+2 x^{4}}$

Determine P(E|F) : A coin is tossed three times.
E : heads on third toss, F : heads on first two tosses.

If $\text{A}=\begin{bmatrix}3&1\\-1&2\end{bmatrix}$ and $\text{I}=\begin{bmatrix}1&0\\0&1\end{bmatrix},$ then find $\lambda$ so that $\text{A}^2 = 5\text{A} + \lambda\text{I}.$

Find the points o local maxima or local minima, if any, of the following functions, using the first derivatives test. Also, find the local maximum or local minimum values, as the case may be:
$\text{f}(\text{x})=\text{x}^{3}-6\text{x}^{2}+9\text{x}+15$

A card is drawn at random from a well-shuffled pack of 52 cards. Find the probability that it is neither an ace nor a king.

Find the volume of the parallelopiped whose coterminous edges are represented by the vectors:
$\vec{\text{a}}=2\hat{\text{i}}-3\hat{\text{j}}+4\hat{\text{k}},\vec{\text{b}}=\hat{\text{i}}+2\hat{\text{j}}-\hat{\text{k}},\vec{\text{c}}=3\hat{\text{i}}-\hat{\text{j}}-2\hat{\text{k}}$

Solve the following differential equation:$4\frac{\text{dy}}{\text{dx}}+\text{8y}=\text{5e}^{-3x}$.

Let $A = R_0 \times R,$ where $R_0$ denote the set of all non$-$zero real numbers. A binary operation $'⊙\ '$ is defined on $A$ as follows: $(a, b) ⊙ (c, d) = (ac, bc + d)$ for all $(a, b), (c, d) \in R_0 \times R.$ Find the invertible elements in $A.$