Download App

Indefinite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals4 Marks
Question
Evaluate the following intregals:
$\int\frac{\text{x}+2}{\sqrt{\text{x}^2+2\text{x}-1}}\text{dx}$
✓

Answer

Let $\text{I}=\int\frac{\text{x}+2}{\sqrt{\text{x}^2+2\text{x}-1}}\text{dx}$
consider,
$\text{x}+2=\text{A}\frac{\text{d}}{\text{dx}}(\text{x}^2+2\text{x}-1)+\text{B}$
$\Rightarrow\text{x}+2=\text{A}(2\text{x}+2)+\text{B}$
$\Rightarrow\text{x}+2=(2\text{A})\text{x}+2\text{A}+\text{B}$
Equating coefficient of like terms.
$2\text{A}=1$
$\Rightarrow\text{A}=\frac{1}{2}$
And
$2\text{A}+\text{B}=2$
$\Rightarrow2\times\frac{1}{2}+\text{B}=2$
$\Rightarrow\text{B}=1$
Then,
$\text{I}=\int\frac{\big[\frac{1}{2}(2\text{x}+2)+1}{\sqrt{\text{x}^2+2\text{x}+1}}\text{dx}$
$=\frac{1}{2}\int\frac{(2\text{x}+2)\text{dx}}{\sqrt{\text{x}^2+2\text{x}+1}}+\int\frac{\text{dx}}{\sqrt{\text{x}^2+2\text{x}-1}}$
Let $\text{x}^2+2\text{x}-1=\text{t}$
$\Rightarrow(2\text{x}+2)\text{dx}=\text{dt}$
$\therefore\text{I}=\frac{1}{2}\int\frac{\text{dt}}{\sqrt{\text{t}}}+\int\frac{\text{dx}}{\sqrt{\text{x}^2+2\text{x}+1-2}}$
$=\frac{1}{2}\int\text{t}^{-\frac{1}{2}}\text{dt}+\int\frac{\text{dx}}{\sqrt{\text{x}^2+2\text{x}+1-2}}$
$=\frac{1}{2}\bigg[\frac{\text{t}^{\frac{1}{2}}+1}{-\frac{1}{2}+1}\bigg]+\int\frac{\text{dx}}{\sqrt{(\text {x}+1)^2-(\sqrt{2})^2}}$
$=\sqrt{\text{t}}+\log\Big|\text{x}+1+\sqrt{(\text{x}+1)^2-(\sqrt{2})^2}\Big|+\text{C}$
$=\sqrt{\text{x}^2+2\text{x}-1}+\log\big|\text{x}+1+\sqrt{\text{x}^2+2\text{x}-1}\big|+\text{C}$

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

Similar questions

Find the area bounded by the cutve y = 4 - x2 and the line y = 0, y = 3.
→
If $\text{x}=\cos\text{t}(3-2\cos^2\text{t}),\text{y}\sin\text{t}(3-2\sin^2\text{t})$ find the value of $\frac{\text{dy}}{\text{dx}}\text{ at t}=\frac{\pi}{4}$
→
If $\text{A}=\begin{bmatrix}-2\\4\\5\end{bmatrix},\text{B}=\begin{bmatrix}1&3&-6\end{bmatrix},$ verify that (AB)T = BTAT
→
Evaluate the following integrals:
$\int^\limits{\frac{\pi}{2}}_0\cos^5\text{x dx}$
→
Three urns A, B and C contain 6 red and 4 white; 2 red and 6 white; and 1 red and 5 white balls respectively. An urn is chosen at random and a ball is drawn. If the ball drawn is found to be red, find the probability that the ball was drawn from urn A.
→
Prove that the function $\text{f(x)}=\begin{cases}\frac{\sin\text{x}}{\text{x}},&\text{x}<0\\\text{x}+1,&\text{x}\geq0\end{cases}$ is everywhere continuous.
→
Evaluate the following integrals:
$\int\limits^{\text{a}}_{-\text{a}}\log\Big(\frac{\text{a}-\sin\theta}{\text{a}+\sin\theta}\Big)\text{d}\theta$
→
Evaluate the following integrals:
$\int\limits^{\frac{\pi}{2}}_0\frac{\sin^{\text{n}}\text{x}}{\sin^\text{n}\text{x}+\cos^\text{n}\text{x}}\text{ dx},\text{ n}\in\text{N}$
→
The coordinates of the foot of the perpendicular drawn from the origin to a plane are (12, -4, 3). Find the equation of the plane.
→
Find the angle between the pairs of lines with direction ratios proportional to

1, 2, -2 and -2, 2, 1

→