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

Question

Evaluate the following integrals:
$\int\Big(\frac{\text{x}+1}{\text{x}}\Big)(\text{x}+\log\text{x})^2\text{dx}$

✓

Answer

$\int\Big(\frac{\text{x}+1}{\text{x}}\Big)(\text{x}+\log\text{x})^2\text{dx}$ Let $\text{x}+\log\text{x}=\text{t}$ $\Rightarrow\Big(1+\frac{1}{\text{x}}\Big)=\frac{\text{dt}}{\text{dx}}$ $\Rightarrow\Big(\frac{\text{x}+1}{\text{x}}\Big)\text{dx}=\text{dt}$Now, $\int\Big(\frac{\text{x}+1}{\text{x}}\Big)(\text{x}+\log\text{x})^2\text{dx}$
$=\int\text{t}^2\text{dt}$
$=\frac{\text{t}^2}{3}+\text{C}$
$=\frac{(\text{x}+\log\text{x})^3}{3}+\text{C}$

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

Similar questions

A die is rolled. If the outcome is an odd number, what is the probability that it is prime?

Prove the following results:
$\sin^{-1}\frac{63}{65}=\sin^{-1}\frac{5}{13}+\cos^{-1}\frac{3}{5}$

Evaluate the following integrals:
$\int\cos\text{x}\sqrt{4-\sin^2\text{x}}\text{dx}$

On the set Z of integers, if the binary operation * is defined by a * b = a + b + 2, then find the identity element.

Let A = [-1, 1]. Then, discuss whether the following functions from A to itself are one-one, onto or bijective:
g(x) = |x|

An urn contains $10$ white and $3$ black balls. Another urn contains $3$ white and $5$ black balls. Two are drawn from first urn and put into the second urn and then a ball is drawn from the latter. Find the probability that its is a white ball.

Check the commutativity and associativity of the following binary operations:
'*' on Q defined by $\text{a}\ ^*\ \text{b}=\frac{\text{ab}}{4}$ for all a, b ∈ Q.

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point $'c'$ in the indicated interval as stated by the Lagrange's mean value theorem.
$f(x) = x^2 - 3x + 2 $ on $[-1, 2]$

Verify that y = A cos x - b sin x is a solution of the differential equation.
$\frac{\text{d}^{2} \text{y}}{\text{dx}^{2}}+\text{y}=0.$

Find a vector $\vec{\text{r}}$ of magnitude $3\sqrt{2}$ units which makes an angle of $\frac{\pi}{4}$ and $\frac{\pi}{2}$ with and z-axes respectively.