Evalute the following integrals: 1 x(3+ ) dx - Vidyadip

Question

Evalute the following integrals:
$\int\frac{1}{\text{x}(3+\log\text{x})}\text{dx}$

✓

Answer

Here, we are considering $\log\text{x}$ as $\log_\text{e}\text{x}$.

Let $\text{I}=\int\frac{1}{\text{x}(3+\log\text{x})}\text{dx}$

Putting $\log\tan\text{x}=\text{t}$

$\Rightarrow\frac{1}{\text{x}}=\frac{\text{dt}}{\text{dx}}$

$\Rightarrow\frac{\text{dx}}{\text{x}}=\text{dt}$

$\therefore\text{I}=\int\frac{\text{dt}}{3+\text{t}}$

$=\log|3+\text{t}|+\text{C}$

$=\log|3+\log\text{x}|+\text{C}\ \big[\because\text{t}=\log\text{x}\big]$

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" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

If $\text{y}=\log(\sin\text{x})$ Prove that $\frac{\text{d}^3\text{y}}{\text{dx}^3}=2\cos\text{x}\ \text{cosec}^3\text{x}$

Find $\frac{\text{dy}}{\text{dx}}$ in the following cases:
$\frac{\text{x}^2}{\text{a}^2}+\frac{\text{y}^2}{\text{b}^2}=1$

Assume that the chances of a patient having a heart attack is 40%. It is also assumed that a meditation and yoga course reduce the risk of heart attack by 30% and prescription of certain drug reduces its chances by 25%. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga?

Find the local maxima and local minima of the function
f(x) = x³ - 6x² + 9x + 15
Find also the local maximum and the local minimum value.

Evaluate the following:
$\tan^{-1}\Big(-\frac{1}{\sqrt3}\Big)+\tan^{-1}\Big(-\sqrt3\Big)+\tan^{-1}\Big(\sin\Big(-\frac{\pi}{2}\Big)\Big)$

Using vector method prove that in any triangle ABC,
$
\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c} .
$

If f : {5, 6} → {2, 3} and g : {2, 3} → {5, 6} are given by f = {(5, 2), (6, 3)} and g = {(2, 5), (3, 6)}, then find fog.

Evaluate the following as limit of sums:
$\int\limits^2_0\text{e}^\text{x}\text{dx}$

Let $\vec{\text{a}}=\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}},\vec{\text{b}}=\hat{\text{i}}$ and $\vec{\text{c}}=\text{c}_1\hat{\text{i}}+\text{c}_2\hat{\text{j}}+\text{c}_3\hat{\text{k}}.$ Then,
If c₁= 1 and c₂= 2, find c₃which makes $\vec{\text{a}},\vec{\text{b}}$ and $\vec{\text{c}}$ coplanar.

State with reasons whether the following functions have inverse:
f : {1, 2, 3, 4} → {10} with f = {(1, 10), (2, 10), (3, 10), (4, 10)}