Download App

Definite Integrals — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSDefinite Integrals5 Marks
Question
Evaluate the following integrals:
$\int\limits^{7}_0\frac{\sqrt[3]{\text{x}}}{\sqrt[3]{\text{x}}+\sqrt[3]{7}-\text{x}}\text{ dx}$
✓

Answer

Let $\text{I}=\int\limits^{7}_0\frac{\sqrt[3]{\text{x}}}{\sqrt[3]{\text{x}}+\sqrt[3]{7-\text{x}}}\text{ dx}\ ...(\text{i})$
We know that $\int\limits^{\text{a}}_0\text{f(x)}=\int\limits^{\text{a}}_0\text{f}(\text{a}-\text{x})$
Hence,
$\text{I}=\int\limits^{7}_0\frac{\sqrt[3]{7-\text{x}}}{\sqrt[3]{7-\text{x}}+\sqrt[3]{\text{x}}}\text{ dx}\ ...(\text{ii})$
Adding (i) & (ii)
$2\text{I}=\int\limits^{7}_0\frac{\sqrt[3]{\text{x}}}{\sqrt[3]{\text{x}}+\sqrt[3]{7-\text{x}}}\text{ dx}+\frac{\sqrt[3]{7-\text{x}}}{\sqrt[3]{7-\text{x}}+\sqrt[3]{\text{x}}}\text{ dx}$
$2\text{I}=\int\limits^{7}_0\frac{\sqrt[3]{\text{x}}+\sqrt[3]{7-\text{x}}}{\sqrt[3]{\text{x}}+\sqrt[3]{7-\text{x}}}\text{ dx}$
$2\text{I}=\int\limits^{7}_0\text{dx}$
$2\text{I}=\big[\text{x}\big]^7_0$
$\text{I}=\frac{7}{2}$

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 "5 Marks Questions" from Definite Integrals→Definite Integrals — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Find the points on the curve $xy + 4 = 0$ at which the tangents are inclined at an angle of $45^\circ$ with the $x-$axis.
→
Differentiate the following functions with respect to x:
$\sin^{-1}\big\{\sqrt{1-\text{x}^2}\big\},0<\text{x}<1$
→
Solve the following equation for x:
$\cot^{-1}\text{x}-\cot^{-1}(\text{x}+2)=\frac{\pi}{12},\text{x}>0$
→
Solve the Linear Programming Problem graphically:
Maximize Z = 3x + 2y subject to $x + 2y \leq 10,3x + y \leq 15,x,y \geq0$
→
The contents of urns $I, II, III$ are as follows:
Urn$ I : 1$ white, $2$ black and $3$ red balls
Urn $II : 2$ white, $1$ black and $1$ red balls
Urn $III : 4$ white, $5$ black and $3$ red balls.
One urn is chosen at random and two balls are drawn. They happen to be white and red. What is the probability that they come from Urns $I, II, III$?
→
Evaluate the follwing intregals:
$\int\frac{1}{\text{x}^4-1}\text{ dx}$
→
Let $\vec{\text{a }}=\hat{\text{i}}+4\hat{\text{j}}+2\hat{\text{k}},\vec{\text{b }}=3\hat{\text{i}}-2\hat{\text{j}}+7\hat{\text{k}}\text{ and }\vec{\text{ c}}=2\hat{\text{i}}-\hat{\text{j}}+4\hat{\text{k}}.$ Find a vector $\vec{\text{ p}}$  which is perpendicular to both $\vec{\text{a }}$ and $\vec{\text{ b}}$ and $\vec{\text{ p}}\cdot\vec{\text{ c}}$ = 18.
→
Evaluate the following integrals:
$\int\text{e}^{2\text{x}}\cos(3\text{x}+4)\text{dx}$
→
Find the particular solution of the differential equation $\frac{\text{dy}}{\text{d}x}=\frac{x(2\log x +1)}{\sin y+y\cos y}$ given that $\text{y}=\frac{\pi}{2}\text{ when } x=1.$
→
Evaluate the following integrals:$\int\frac{\text{x}^3\sin^{-1}\text{x}^2}{\sqrt{1-\text{x}^4}}\text{dx}$
→