Evaluate _ 1 ^ 2 x d x (x+1)(x+2) - Vidyadip

Question

Evaluate $\int_{1}^{2} \frac{x d x}{(x+1)(x+2)}$

✓

Answer

Let $I=\int_{1}^{2} \frac{x d x}{(x+1)(x+2)}$
Using partial fraction, we get $\frac{x}{(x+1)(x+2)}=\frac{-1}{x+1}+\frac{2}{x+2}$
So, $\int \frac{x d x}{(x+1)(x+2)}$ = - log |x + 1| + 2log |x + 2| = F(x)
Therefore, by the second fundamental theorem of calculus, we have
I = F(2) - F(1) = [- log 3 + 2 log 4] - [- log 2 + 2 log 3]
= -3 log 3 + log 2 + 2 log 4 = log$ \left(\frac{32}{27}\right)$

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 "1 Marks Question" from Integrals→Integrals — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

If $\text{A}[\text{a}_{\text{ij}}]=\begin{bmatrix}2&3&-5\\1&4&9\\0&7&-2\end{bmatrix}$ and $\text{B}=[\text{b}_\text{ij}]=\begin{bmatrix}2&-1\\-3&4\\1&-2\end{bmatrix}$ Then find $a_{22} + b_{21}$

Show that the function $f: R \to R,$ defined as $f(x) = x^2 ,$ is neither one$-$one nor onto.

The binary operation * : R x R $\rightarrow$ R is defined as a * b = 2a + b. Find (2 * 3) * 4.

Evaluate the following:
$\cos^{-1}(\cos3)$

Construct a $3 \times 4$ matrix $A = [a_{ij}]$ whose element $a_{ij}$ are given by: $a_{ij} = j$

Write the vector equation of a line given by $\frac{\text{x - 5}}{3}=\frac{\text{y + 4}}{7}=\frac{\text{z - 6}}{2}.$

Write the principal value of $\tan^{-1} (1)+ \cos^{-1}\bigg(- \frac{1}{2}\bigg).$

Integrate the function $x \tan^{-1}x$

Define position vector.

Integrate the functions in Exercises:
$\frac{\sin^{-1}\text{x}}{\sqrt{1-\text{x}^2}}$