Download App

Definite Integration — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDefinite Integration3 Marks
Question
Evaluate : $\int_1^2 \frac{\log x}{x^2} \cdot d x$
✓

Answer

$
\begin{aligned}
& \text { Let I }=\int_1^2(\log x)\left(\frac{1}{x^2}\right) \cdot d x \\
&=\left[(\log x) \cdot \int \frac{1}{x^2} \cdot d x\right]_1^2-\int_1^2 \frac{d}{d x} \log x \cdot \int \frac{1}{x^2} \cdot d x \cdot d x \\
&=\left[(\log x) \cdot\left(-\frac{1}{x}\right)\right]_1^2-\int_1^2 \frac{1}{x} \cdot\left(-\frac{1}{x}\right) \cdot d x \\
&=\left[-\frac{1}{x} \log x\right]_1^2+\int_1^2 \frac{1}{x^2} \cdot d x \\
&=\left[-\frac{1}{x} \log x\right]_1^2+\left[-\frac{1}{x}\right]_1^2 \\
&=\left[\left(-\frac{1}{2} \log 2\right)-\left(-\frac{1}{1} \log 1\right)\right]+\left[\left(-\frac{1}{2}\right)-\left(-\frac{1}{1}\right)\right] \\
&=-\frac{1}{2} \log 2-0-\frac{1}{2}+1=\frac{1}{2}-\frac{1}{2} \log 2 \\
& \therefore \quad \int_1^2 \frac{1}{x^2}\left(\frac{\log x}{2}(1-\log 2)\right.
\end{aligned}
$

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 "Solve the Following Question.(3 Marks)" from Definite Integration→Definite Integration — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Two cards are drawn successively without replacement from a well-shuffled deck of 52 cards. Find the probability of exactly one ace.
→
If $2 \tan ^{-1}(\cos x)=\tan ^{-1}(\operatorname{cosec} x)$ then find the value of $x$.
→
A speaks the truth 8 times out of 10 times. A die is tossed. He reports that it was 5. What is the probability that it was actually 5?
→
Let $\hat{a}, \hat{b}, \hat{c}$ be unit vectors such that $\hat{a} \cdot \hat{b}=\hat{a} \cdot \hat{c}=0$ and the angle between $\hat{b}$ and $\hat{c}$ be

$\frac{\pi}{6}$.

Prove that $\hat{a}= \pm 2(\hat{b} \times \hat{c})$

→
Evaluate the following integrals:$\int\frac{\log\text{x}}{\text{x}^{\text{n}}}\text{dx}$
→
Find the approximate value of $e^{1.005}$. Given that $e=2.7183$..
→
Without using truth table show that : $\sim(p \vee q) \vee(\sim p \wedge q) \equiv \sim p$
→
Evaluate the following integrals:$\int\limits^{\text{a}}_0\frac{\sqrt{\text{x}}}{\sqrt{\text{x}}+\sqrt{\text{a}-\text{x}}}\text{ dx}$
→
Evaluate: $\int \frac{1}{3+5 \cos x} d x$
→
Find the inverse of each of the following matrices $($if they exist$) : \left[\begin{array}{cc}3 & -10 \\ 2 & -7\end{array}\right]$
→