Evaluate the following integrals: ^ 8 _2 10-x x + 10-x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits^{8}_2\frac{\sqrt{10-\text{x}}}{\sqrt{\text{x}}+\sqrt{10-\text{x}}}\text{ dx}$

✓

Answer

Let $\text{I}=\int\limits^{8}_2\frac{\sqrt{10-\text{x}}}{\sqrt{\text{x}}+\sqrt{10-\text{x}}}\text{ dx}\ ...(\text{i})$
Then,
$\text{I}=\int\limits^{8}_2\frac{\sqrt{10-(2+8-\text{x}})}{\sqrt{2+8-\text{x}}+\sqrt{10-(2+8-\text{x}})}\text{ dx}$ $\Bigg[\int\limits^{\text{b}}_\text{a}\text{f(x)}\text{dx}=\int\limits^{\text{b}}_\text{a}\text{f}(\text{a}+\text{b}-\text{x})\text{dx}\Bigg]$
$=\int\limits^{8}_2\frac{\sqrt{\text{x}}}{\sqrt{10-\text{x}}+\sqrt{\text{x}}}\text{ dx}\ ...(\text{ii})$
Adding (i) and (ii) we get
$2\text{I}=\int\limits^{8}_2\frac{\sqrt{10-\text{x}}}{\sqrt{\text{x}}+\sqrt{10-\text{x}}}\text{ dx}$
$\Rightarrow2\text{I}=\int\limits^{8}_2\text{dx}$
$\Rightarrow2\text{I}=\big[\text{x}\big]^8_2$
$\Rightarrow2\text{I}=8-2=6$
$\Rightarrow\text{I}=3$

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

Similar questions

Using vectors find the area of the triangle with vertices, A(2, 3, 5), B(3, 5, 8) and C(2, 7, 8).

Find $\frac{\text{dy}}{\text{dx}}$ in the following cases:
$(x^2 + y^2)^2 = xy$

Solve the differential equation $\frac{\text{dy}}{\text{dx}}+2\text{xy}=\text{y}.$

A firm manufactures two products $A$ and $B$. Each product is processed on two machines $M_1$ and $M_2$. Product $A$ requires $4$ minutes of processing time on $M_1$ and $8$ min. on $M_2;$ product $B$ requires $4$ minutes on $M_1$ and $4$ min. on $M_2$. The machine $M_1$ is available for not more than $8$ hrs $20$ min. while machine $M_2$ is available for $10$ hrs. during any working day. The products $A$ and $B$ are sold at a profit of Rs. $3$ and Rs. $4$ respectively.
Formulate the problem as a linear programming problem and find how many products of each type should be produced by the firm each day in order to get maximum profit.

Without expanding, show that the values of the following determinant are zero:
$\begin{vmatrix}\frac{1}{\text{a}}&\text{a}^2&\text{bc}\\\frac{1}{\text{b}}&\text{b}^2&\text{ac}\\\frac{1}{\text{c}}&\text{c}^2&\text{ab} \end{vmatrix}$

Find one-parameter families of solution curves of the following differential equation: (or solve the following differential equation)$\text{x}\frac{\text{dy}}{\text{dx}}+2\text{y}=\text{x}^2\log\text{x}$

Find the equation of the plane through the line of intersection of the planes $\vec{\text{r}}\cdot(\hat{\text{i}}+3\hat{\text{j}})+6=0$ and $\vec{\text{r}}\cdot(3\hat{\text{i}}-\hat{\text{j}}-4\hat{\text{k}})=0,$ which is at a unit distance from the origin.

Show that the semi $-$ vertical angle of a cone of maximum volume and given slant height is $\tan ^{-1} \sqrt{2}$ or $\cos ^{-1} \frac{1}{\sqrt{3}}$.

If $\text{A}=\begin{bmatrix}2 & -3 & 5 \\3 & 2 & -4\\1 & 1 & -2 \end{bmatrix},$ then find $A^{–1}$. Hence solve the following system of equations: $2x - 3y + 5z = 11, 3x + 2y - 4z = -5, x + y - 2z = -3$.

Find the angle between the line joining the points (3, -4, -2) and (12, 2, 0) and the plane 3x - y + z = 1.