Evaluate the following definite integrals: _ 0 ^1 x x+1 dx - Vidyadip

Question

Evaluate the following definite integrals:$\int_{0}^\limits{1}\frac{\text{x}}{\text{x}+1}\text{ dx}$

✓

Answer

Let $\text{I}=\int_{0}^\limits{1}\frac{\text{x}}{\text{x}+1}\text{ dx}$ Then,$\text{I}=\int_{0}^\limits{1}1-\frac{1}{\text{x}+1}\text{ dx}$
$\Rightarrow\text{I}=\big[\text{x}-\log(\text{x}+1)\big]_0^1$
$\Rightarrow\text{I}=1-\log2-(0-\log1)$
$\Rightarrow\text{I}=\log\text{e}-\log2$
$\Rightarrow\text{I}=\log\frac{\text{e}}{2}$

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.(2 Marks)" from Definite Integrals→Definite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Prove the following results:
$\tan^{-1}\frac{1}{7}+\tan^{-1}\frac{1}{13}=\tan^{-1}\frac{2}{9}$

$\sim[p \rightarrow(p \wedge \sim q)] \equiv$

If $f: C \rightarrow C$ is defined by $f(x)=(x-2)^3$, write $f^{-1}(-1)$.

Find an angle $\theta$
Whose rate of increase twice is twice the rate of decrease of its cosine.

Find the angle between the planes $\bar{r} \cdot(2 \bar{i}+\bar{j}-\bar{k})=3$ and $\bar{r} \cdot(\bar{i}+2 \bar{j}+\bar{k})=1$

Evaluate: $\int_0^{\frac{\pi}{4}} \frac{\cos x}{4-\sin ^2 x} d x$

If $\text{A}=\begin{bmatrix}1&1\\1&1\end{bmatrix}$ satisfies $\text{A}^4=\lambda\text{A},$ then write the value of $\lambda.$

If the vectors $-3 \hat{i}+4 \hat{j}-2 \hat{k}, \hat{i}+2 \hat{k}$ and $\hat{i}-p \hat{j}$ are coplanar, then find the value of $p$.

If the probability distribution of a random variable X is given below:

$X = X_i$	1	2	3	4
$P(X = X_i)$	c	2c	4c	4c

Write the value of $\text{P}(\text{X}\leq2)$

Is it true that every relation which is symmetric and transitive is also reflexive? Give reasons.