Download App

Definite Integration — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDefinite Integration3 Marks
Question
Evaluate : $\int_1^3 \frac{1}{\sqrt{2+x}+\sqrt{x}} \cdot d x$
✓

Answer

$\begin{aligned} & =\int_1^3\left(\frac{1}{\sqrt{2+x}+\sqrt{x}}\right)\left(\frac{\sqrt{2+x}-\sqrt{x}}{\sqrt{2+x}-\sqrt{x}}\right) \cdot d x \\ & =\int_1^3\left(\frac{\sqrt{2+x}-\sqrt{x}}{2+x-x}\right) \cdot d x \\ & =\frac{1}{2} \cdot \int_1^3(\sqrt{2+x}-\sqrt{x}) \cdot d x \\ & =\frac{1}{2} \cdot\left[\frac{(2+x)^{\frac{3}{2}}}{\frac{3}{2}}-\frac{x^{\frac{3}{2}}}{\frac{3}{2}}\right]_1^3\end{aligned}$
$\begin{aligned} & =\frac{1}{3} \cdot\left[(2+x)^{\frac{3}{2}}-(x)^{\frac{3}{2}}\right]_1^3 \\ & =\frac{1}{3}\left\{\left[(2+3)^{\frac{3}{2}}-(3)^{\frac{3}{2}}\right]-\left[(2+1)^{\frac{3}{2}}-(1)^{\frac{3}{2}}\right]\right\} \\ & =\frac{1}{3}\left\{5^{\frac{3}{2}}-3^{\frac{3}{2}}-3^{\frac{3}{2}}+1^{\frac{3}{2}}\right\} \\ & =\frac{1}{3}\left\{5^{\frac{3}{2}}-2(3)^{\frac{3}{2}}+1\right\} \\ & \therefore \int_1^3 \frac{1}{\sqrt{2+x}+\sqrt{x}} d x=\frac{1}{3}\left[5^{\frac{3}{2}}-2(3)^{\frac{3}{2}}+1\right]\end{aligned}$

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 Integration→Definite Integration — 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 co-ordinates of the foot of the perpendicular drawn from the origin to the plane $2 x+y-2 z=18$.
→
Solve the following differential equations:
$\sqrt{1+\text{x}^2}\ \text{dy}+\sqrt{1+\text{y}^2}\ \text{dx}=0$
→
Solve: $\cos\big(\sin^{-1}\text{x}\big)=\frac{1}{6}$
→
For any two vectors $\vec{\text{a}}$ and $\vec{\text{b}}$ of magnitudes 3 and 4 respectively, write the value of $\big[\vec{\text{a}}\vec{\text{b}}\vec{\text{a}}\times\vec{\text{b}}\big]+\big(\vec{\text{a}}.\vec{\text{b}}\big)^2$
→
Differentiate $\tan ^{-1}\left(\frac{\cos x}{1+\sin x}\right)$ w.r.t. $\sec ^{-1} x$.
→
A coin is tossed three times. Find $\text{P}\Big(\frac{\text{A}}{\text{B}}\Big)$ in each of the following:
A = At least two heads,
B = At most two heads.
→
Find the vector equation of the line passing through the point (2, -1, -1) which is parallel to the line 6x - 2 = 3y +1 =2z - 2.
→
In a family, the husband tells a lie in 30% cases and the wife in 35% cases. Find the probability that both contradict each other on the same fact.
→
Let $f(x) = x^2 + x + 1$ and $g(x) = sinx$. Show that fog ≠ gof.
→
Differentiate:$\tan(\text{x}^\circ+45^\circ)$
→