Download App

INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS5 Marks
Question
$\int\text{e}^{\tan^{-1}\text{x}}\Big(\frac{1+\text{x}+\text{x}^2}{1+\text{x}^2}\Big)\text{dx}$
✓

Answer

Let $\text{I}=\int\text{e}^{\tan^{-1}\text{x}}\Big(\frac{1+\text{x}+\text{x}^2}{1+\text{x}^2}\Big)\text{dx}$
$\int\text{e}^{\tan^{-1}\text{x}}\bigg(\frac{1+\text{x}^2}{1+\text{x}^2}+\frac{\text{x}}{1+\text{x}^2}\bigg)\text{dx}$
$\int\text{e}^{\tan^{-1}\text{x}}\text{dx}+\int\frac{\text{x e}^{\tan^{-1}\text{x}}}{1+\text{x}^2}\text{dx}$
$\text{I}=\text{I}_1+\text{I}_2\ \ \dots(\text{i})$
Now, $\text{I}_2=\int\frac{\text{x e}^{\tan^{-1}\text{x}}}{1+\text{x}^2}\text{dx}$
Put $\tan^{-1}\text{x}=\text{t}\Rightarrow\text{x}=\tan\text{t}$
$\Rightarrow\ \frac{1}{1+\text{x}^2}\text{dx}=\text{dt}$
$\therefore\ \text{I}=\int\tan\text{t}\cdot\text{e}^\text{t}\text{dt}\\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{I }\ \ \ \ \ \ \ \ \ \text{II}$
$=\tan\text{t}\cdot\text{e}^\text{t}-\int\sec^2\text{t}\cdot\text{e}^\text{t}\text{dt}+\text{C}$
$=\tan\text{t}\cdot\text{e}^\text{t}-\int(1+\tan^2\text{t})\text{e}^\text{t}\text{dt}+\text{C}$ $[\because\sec^2\theta=1+\tan^2\theta]$
$\text{I}_2=\tan\text{t}\cdot\text{e}^\text{t}-\int(1+\text{x}^2)\frac{\text{e}^{\tan^{-1}\text{x}}}{1+\text{x}^2}\text{dx}+\text{C}$
$\text{I}_2=\tan\text{t}\cdot\text{e}^\text{t}-\int\text{e}^{\tan^{-1}\text{x}}\text{dx}+\text{C}$
$\therefore\ \text{I}=\int\text{e}^{\tan^{-1}\text{x}}\text{dx}+\tan\text{t}\cdot\text{e}^\text{t}-\int\text{e}^{\tan^{-1}\text{x}}\text{dx}+\text{C}$
$=\tan\text{t}\cdot\text{e}^\text{t}+\text{C}$
$=\text{x}\text{ e}^{\tan^{-1}\text{x}}+\text{C}$

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 "5 Marks Questions" 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

A small firm manufactures necklaces and bracelets. The total number of necklaces and bracelets that it can handle per day is at most 24. It takes one hour to make a bracelet and half an hour to make a necklace. The maximum number of hours available per day is 16. If the profit on a necklace is Rs. 100 and that on a bracelet is Rs. 300. Formulate on L.P.P. for finding how many of each should be produced daily to maximize the profit?
It is being given that at least one of each must be produced.
→
$$Find the inverse of each of the matrix:
$\begin{bmatrix}1&0&0\\0&\cos\alpha&\sin\alpha\\0&\sin\alpha&-\cos\alpha\end{bmatrix}$
→
Find the shortest distance between lines $\vec{\text{r}}=6\hat{\text{i}}+2\hat{\text{j}}+2\hat{\text{k}}+\lambda\Big(\hat{\text{i}}-2\hat{\text{j}}+2\hat{\text{k}}\Big)\ \text{and}\ \vec{\text{r}}=-4\hat{\text{i}}-\hat{\text{k}}+\mu\Big(3\hat{\text{i}}-2\hat{\text{j}}-2\hat{\text{k}}\Big).$
→
Find the particular solution of the following differential equation;
$\frac{\text{dx}}{\text{dy}} = 1 + x^2 + y^2 + x^2y^2, $ given that $y = 1$ when $x = 0$.
→
A dealer in rural area wishes to purchase a number of sewing machines. He has only ₹ 5,760 to invest and has space for at most 20 items for storage. An electronic sewing machine costs him ₹ 360 and a manually operated sewing machine ₹ 240. He can sell the sewing machine at a profit of ₹ 22 and a manually operated sewing machine at a profit of ₹ 18. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize his profit? Make it as an LPP and solve it graphically.
→
The probability that a certain kind of component will survive a given shock test is $\frac{3}{4}.$ Find the probability that among 5 components tested.
  1. exactly 2 will survive.
  2. at most 3 will survive.
→
$\text{Let } \vec{\text a} = \hat{\text{i}} + \hat{\text{j}} + \hat{\text{k}}, \vec{\text{b}} = \hat{\text{i}} \text{ and } \vec{\text{c}} = \text{c}_{1} \hat{\text{i}} + \text{c}_{2} \hat{\text{j}} + \text{c}_{3} \hat{\text{k}}, \text{then}$
  1. Let $c_1 = 1$ and $c_2 = 2$, find $c_3$ which makes $\vec{\text{a}}, \vec{\text{b}} \text{ and }\vec{\text{c}} \text{ coplanar.}$
  2. If $c_2 = –1$ and $c_3 = 1$, show that no value of $c_1$ can make $\vec{\text{a}}, \vec{\text{b}} \text{ and } \vec{\text{c}} \text{ coplanar}.$
→
$\int\frac{\text{x}^2+3\text{x}+1}{(\text{x}+1)^2}\text{dx}$
→
Let $f : N \rightarrow N$ be a function as $f(x) = 9x^2 + 6x - 5$. Show that $f : N \rightarrow S,$ where S is the range of $f,$ is invertible. Find the inverse of $f$ and hence find $f^{-1}(43)$ and $f^{-1}(163).$
→
Using definite intergeals, find the area of the region bounded by the following curves, after making a rough sketch y = 1 + |x + 1|, x = -2, y = 0.
→