Evaluate the following integrals: 1 1+ x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\frac{1}{1+\sqrt{\text{x}}}\text{dx}$

✓

Answer

Let $\text{I}=\int\frac{1}{1+\sqrt{\text{x}}}\text{dx}\ \ ....(1)$
Let $\text{x}=\text{t}^2$ then,
$\text{dx}=\text{d}(\text{t}^2)$
$\Rightarrow\text{dx}=2\text{t}\text{ dt}$
Putting $\text{x}=\text{t}^2$ and $\text{dx}=2\text{t}\text{ dt}$ in equation (1), we get
$\text{I}=\int\frac{2\text{t}}{1+\sqrt{\text{t}^2}}\text{dt}$
$=\int\frac{2\text{t}}{1+\text{t}}\text{dt} $
$=2\int\frac{\text{t}}{1+\text{t}}\text{dt}$
$=2\int\frac{1+\text{t}-1}{1+\text{t}}\text{dt}$
$=2\int\Big[\frac{1+\text{t}}{1+\text{t}}-\frac{1}{1+\text{t}}\Big]\text{dt}$
$=2\int\text{dt}-2\int\frac{1}{1+\text{t}}\text{dt} $
$=2\text{t}-2\log(1+\text{t})+\text{C}$
$=2\sqrt{\text{x}}-2\log(1+\sqrt{\text{x}})+\text{C}$
$\therefore\text{ I}=2\sqrt{\text{x}}-2\log(1+\sqrt{\text{x}})+\text{C}$

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 "Solve the Following Question.(5 Marks)" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Evaluate the following integrals:$\int\frac{\text{x}^2+1}{\text{x}^2-5\text{x}+6}\text{ dx}$

If w is a complex cube root of unity, show that.
$\begin{pmatrix}\begin{bmatrix}1&w&w^2\\w&w^2&1\\w^2&1&w\end{bmatrix} +\begin{bmatrix}w&w^2&1\\w^2&1&w\\w&w^2&1\end{bmatrix}\end{pmatrix}\begin{bmatrix}1\\w\\w^2\end{bmatrix}=\begin{bmatrix}0\\0\\0\end{bmatrix}$

If $|\vec{\text{a}}|=13,\big|\vec{\text{b}}\big|=5$ and $\vec{\text{a}}.\vec{\text{b}}=60,$ then find $\big|\vec{\text{a}}\times\vec{\text{b}}\big|.$

Find the value of $\lambda$ so that lines $\frac{1-x}{3}=\frac{7 y-14}{2 \lambda}=\frac{z-3}{2}$ and $\frac{7-7 x}{3 \lambda}=\frac{y-5}{1}=\frac{6-z}{5}$ are at right angle.

Find the equation of the curve which passes through the point (2, 2) and satisfies the differential equation $\text{y}-\text{x}\frac{\text{dy}}{\text{dx}}=\text{y}^{2}+\frac{\text{dy}}{\text{dx}}.$

Evaluate the following integrals:$\int_{0}^\limits{1}\tan^{-1}\text{x dx}$

If f is an integrable function such that f(2a - x) = f(x), then prove that:$\int\limits^{2\text{a}}_0\text{f(x)}\text{dx}=2\int\limits^{\text{a}}_0\text{f(x)}\text{dx}$

Solve the following initial value problems:
$\text{x}\frac{\text{dy}}{\text{dx}}-\text{y}=(\text{x}+1)\text{e}^{\text{x}},\text{ y}(1)=0$

If four points A, B, C and D with position vectors $4\hat{\text{i}}+3\hat{\text{j}},5\hat{\text{i}}+\text{x}\hat{\text{j}}+7\hat{\text{k}},5\hat{\text{i}}+3\hat{\text{j}}$ and $7\hat{\text{i}}+6\hat{\text{j}}+\hat{\text{k}}$ respectively are coplanar, then find the value of x.

Find a 2 × 2 matrix A such that.
$\text{A}\begin{bmatrix}1&-2\\1&4\end{bmatrix}=6\text{I}_2$