Download App

Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIntegrals4 Marks
Question
Integrate the function: $\frac{x+2}{\sqrt{x^{2}-1}}$
✓

Answer

Clearly, we can write, x + 2 = $\frac{1}{2}(2 x)+2$
$\Rightarrow \int \frac{x+2}{\sqrt{x^{2}-1}} d x=\int \frac{\frac{1}{2}(2 x)+2}{\sqrt{x^{2}-1}} d x$ 
$= \frac{1}{2} \int \frac{2 \mathrm{x}}{\sqrt{\mathrm{x}^{2}-1}} \mathrm{dx}+2 \int \frac{1}{\sqrt{\mathrm{x}^{2}-1}} \mathrm{d} \mathrm{x}$ 
Now in, $I_1$= $\frac{1}{2} \int \frac{2 x}{\sqrt{x^{2}-1}} d x$ 
Let x2 - 1 = t
$\Rightarrow$ (2x)dx = dt
$\Rightarrow I_1 = \frac{1}{2} \int \frac{2 \mathrm{x}}{\sqrt{\mathrm{x}^{2}-1}} \mathrm{dx}=\frac{1}{2} \int \frac{\mathrm{dt}}{\sqrt{\mathrm{t}}}=\frac{1}{2}[2 \sqrt{\mathrm{t}}]$ = $\frac{1}{2}[2 \sqrt{\mathrm{x^2-1}}]$ 
And $2 \int \frac{1}{\sqrt{x^{2}-1}} d x=2 \log |x+\sqrt{x^{2}-1}|$ 
$\Rightarrow \int \frac{\mathrm{x}+2}{\sqrt{\mathrm{x}^{2}-1}} \mathrm{dx}=\sqrt{\mathrm{x}^{2}-1}+2 \log |\mathrm{x}+\sqrt{\mathrm{x}^{2}-1}|+\mathrm{C}$

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

Similar questions

The cost of 4kg onion, 3kg wheat and 2kg rice is Rs. 60. The cost of 2kg onion, 4kg wheat and 6kg rice is Rs. 90. The cost of 6kg onion 2kg wheat and 3kg rice is Rs. 70. Find the cost of each item per kg by matrix method.
→
David wants to invest at most Rs. 12,000 in Bonds A and B. According to the rule, he has to invest at least Rs. 2,000 in Bond A and at least Rs. 4,000 in Bond B. If the rates of interest on Bonds A and B respectively are 8% and 10% per annum, formulate the problem as L.P.P. and solve it graphically for maximum interest. Also determine the maximum interest received in a year.
→
Differentiate the functions given in Exercise:
$\text{x}^{\sin\text{x}}+(\sin\text{x})^{\cos\text{x}}$
→
Show that if x2 – 5x + 6 = 0, then x = 3 or x = 2
→
Solve the following differential equation
$\text{x}(\text{x}^{2} - 1)\frac{\text{dy}}{\text{dx}} = 1, \text{y}(2) = 0$
→
Find the value of $\lambda$ for which the four points with position vectors
$-\hat{\text{j}}-\hat{\text{k}},4\hat{\text{i}}+5\hat{\text{j}}+\lambda\hat{\text{k}},3\hat{\text{i}}+9\hat{\text{j}}+4\hat{\text{k}}$ and $-4\hat{\text{i}}+4\hat{\text{j}}+4\hat{\text{k}}$ are co planar.
→
A balloon in the form of a right circular cone surmounted by a hemisphere, having a diametre equal to the height of the cone, is being inflated. How fast is its volume changing with respect to its total height h, when h = 9cm.
→
Find the value of x if $\begin{bmatrix}1&\text{x}&1\end{bmatrix}\begin{bmatrix}1&3&2\\2&5&1\\15&3&2\end{bmatrix}\begin{bmatrix}1\\2\\\text{x}\end{bmatrix}=0$
→
Find the distance between the lines l1 and l2 given by

$\vec{\text{r}}=\hat{\text{i}}+2\hat{\text{j}}-4\hat{\text{k}}+\lambda\big(2\hat{\text{i}}+3\hat{\text{j}}+6\hat{\text{k}}\big)$ and, $\vec{\text{r}}=3\hat{\text{i}}+3\hat{\text{j}}-5\hat{\text{k}}+\mu\big(2\hat{\text{i}}+3\hat{\text{j}}+6\hat{\text{k}}\big)$

→
Find the angle between the following pairs of lines:

$\frac{5-\text{x}}{-2}=\frac{\text{y}+3}{1}=\frac{1-\text{z}}{3}$ and $\frac{\text{x}}{3}=\frac{1-\text{y}}{-2}=\frac{\text{z}+5}{-1}$

→