Evaluate the following definite integrals: _ 0 ^11 1+x - x dx - Vidyadip

Question

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

✓

Answer

Let $\text{I}=\int_{0}^\limits{1}\frac{1}{\sqrt{1+\text{x}}-\sqrt{\text{x}}}\text{ dx}$ Then,$\text{I}=\int_{0}^\limits{1}\bigg(\frac{1}{\sqrt{1+\text{x}}-\sqrt{\text{x}}}\times\frac{\sqrt{1+\text{x}}+\sqrt{\text{x}}}{\sqrt{1+\text{x}}+\sqrt{\text{x}}}\bigg)\text{dx}$
$\Rightarrow\text{I}=\int_{0}^\limits{1}\frac{\sqrt{1+\text{x}}-\sqrt{\text{x}}}{1+\text{x}-\text{x}}\text{ dx}$
$\Rightarrow\text{I}=\int_{0}^\limits{1}\Big({\sqrt{1+\text{x}}+\sqrt{\text{x}}}\Big)\text{dx}$
$\Rightarrow\text{I}=\Big[\frac{2}{3}(1+\text{x})^{\frac{3}{2}}+\frac{2}{3}\text{x}^{\frac{3}{2}}\Big]^1_0$
$\Rightarrow\text{I}=\frac{2}{3}\times2\sqrt{2}+\frac{2}{3}-\frac{2}{3}$
$\Rightarrow\text{I}=\frac{4\sqrt{2}}{3}$

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.(3 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

Find the derivative of $\cos ^{-1} x$ w.r. to $\sqrt{1-x^2}$

Find the matrix $X$ such that $\left[\begin{array}{lll}1 & 2 & 3 \\ 2 & 3 & 2 \\ 1 & 2 & 2\end{array}\right] \quad X=\left[\begin{array}{ccc}2 & 2 & -5 \\ -2 & -1 & 4 \\ 1 & 0 & -1\end{array}\right]$

Test the continuity of the function on f(x) at the origin:
$\text{f}\ (\text{x})=\begin{cases}\frac{\text{x}}{\text{|x|}},& \text{x}\neq0\\1, & \text{x} = 0\end{cases}$

Find the principal values of the following:
$\text{cosec}^{-1}\Big(2\cos\frac{2\pi}{3}\Big)$

Differentiate the following functions with respect to x:
$\cos(\log\text{ x})^2$

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

Find the value of $\lambda$ so that the following vectors are coplanar:
$\vec{\text{a}}=2\hat{\text{i}}-\hat{\text{j}}++\hat{\text{k}},\vec{\text{b}}=\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}},\vec{\text{c}}=\lambda\hat{\text{i}}+\lambda\hat{\text{j}}+5\hat{\text{k}}$

Give a condition that three vectors $\vec{\text{a}},\ \vec{\text{b}}\text{ and }\vec{\text{c}}$ from the three sides of a triangle. what are the other possibilities?

Find the principal solutions of the following equations : $\sin 2 \theta=-\frac{1}{2}$

A bag contains 1 red and 2 green balls. One ball is drawn from the bag at random, its colour is noted, and then ball is put back in the bag. One more ball is drawn from the bag at random and its colour is also noted. Let $X$ denote the number of red balls drawn from the bag as described above. Derive the probability distribution of $X$.