Evaluate the following integrals: ^ x ( +1 2 )dx - Vidyadip

Question

Evaluate the following integrals:

$\int\text{e}^{\text{x}}(\log\text{x}+\frac{1}{2})\text{dx}$

✓

Answer

Let $\text{I}=\int\text{e}^{\text{x}}(\log\text{x}+\frac{1}{2})\text{dx}$
Here, $\text{f(x)}=\log\text{x}$
$\Rightarrow\text{f}'\text{(x)}=\frac{1}{\text{x}}$
Put $\text{e}^{\text{x}}\text{f(x)}=\text{t}$
$\Rightarrow\text{e}^{\text{x}}\log\text{x}=\text{t}$
Diff. both sides w.r.t x
$\text{e}^{\text{x}}\log\text{x}+\text{e}^{\text{x}}\frac{1}{\text{x}}=\frac{\text{dt}}{\text{dx}}$
$\Rightarrow\text{e}^{\text{x}}(\log\text{x}+\frac{1}{\text{x}})\text{dx = dt}$
$\therefore\int\text{e}^{\text{x}}\big[\log\text{x}+\frac{1}{\text{x}}\big]\text{dx}=\int\text{dt}$
$\Rightarrow\text{I}=\text{t}+\text{C}$
$=\text{e}^{\text{x}}\log\text{x}+\text{C}$

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

Write the ratio in which the plane 4x + 5y − 3z = 8 divides the line segment joining the points (−2, 1, 5) and (3, 3, 2).

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

Evaluate the following integrals:
$\int^\limits{\frac{\pi}{4}}_{\frac{-\pi}{4}}|\sin\text{x}|\text{dx}$

Let g(x) be the inverse of an invertible function f(x) which is derivable at x = 3. If f(3) = 3 and f(3) = 9, write the value of g'(9).

Evaluate: $\sin^{-1}\Big(\sin\frac{3\pi}{5}\Big).$

Of the students in a college, it is known that 60% reside in a hostel and 40% do not reside in hostel. Previous year results report that 30% of students residing in hostel attain A grade and 20% of ones not residing in hostel attain A grade in their annual examination. At the end of the year, one students is chosen at random from the college and he has an A grade. What is the probability that the selected student is a hosteler?

If y = log_e x, then find $\triangle\text{y}$ when x = 3 and $\triangle\text{x}$ = 0.03.

Differentiate the function $\sqrt {\frac{{\left( {x - 1} \right)\left( {x - 2} \right)}}{{\left( {x - 3} \right)\left( {x - 4} \right)\left( {x - 5} \right)}}} $ w.r.t. x.

Write the set of values of k for which $\text{f}(\text{x})=\text{k}\text{x}-\sin\text{x}$ is increasing on R.

Write a value of $\int\frac{\cos\text{x}}{3+2\sin\text{x}}\text{ dx}$