Download App

Indefinite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals5 Marks
Question
$\int\frac{2-3\text{x}}{\sqrt{1+3\text{x}}}\text{dx}$
✓

Answer

Let I $=\int\frac{2-3​​\text{x}}{\sqrt{1+3\text{x}}}\times\text{dx}.$ Then,
$\text{I}=\int\frac{2-3\text{x}-1+1}{\sqrt{1+3\text{x}}}\text{dx}$
$=\int\frac{-3\text{x}-1+3}{\sqrt{1+3\text{x}}}\text{dx}$
$=\int-\frac{(3\text{x}+1)}{\sqrt{1+3\text{x}}}\text{dx}+3\int\frac{1}{\sqrt{1+3\text{x}}}\text{dx}$
$=-1\int\frac{1+3\text{x}}{\sqrt{1+3\text{x}}}\text{dx}+3\int\frac{1}{\sqrt{1+3\text{x}}}\text{dx}$
$=-1\int(1+3\text{x})^\frac{1}{2}\text{dx}+3\int(1+3\text{x})^\frac{-1}{2}\text{dx}$
$=-1\times\frac{(1+3\text{x})^\frac{3}{2}}{\frac{3}{2}\times3}+3\times\frac{(1+3\text{x})^\frac{1}{2}}{\frac{1}{2}\times3}+\text{C}$
$=-\frac{2}{9}\times(1+3\text{x})^\frac{3}{2}+2(1+3\text{x})^\frac{1}{2}+\text{C}$
$=2(1+3\text{x})^\frac{1}{2}\Big[-\frac{1}{9}(1+3\text{x})^1+1\Big]+\text{C}$
$=2(1+3\text{x})^\frac{1}{2}\Big[\frac{-1-3\text{x}+9}{9}\Big]+\text{C}$
$=2(1+3\text{x})^\frac{1}{2}\Big[\frac{8-3\text{x}}{9}\Big]+\text{C}$
$=\frac{2}{9}\sqrt{1+3\text{x}}(8-3\text{x})+\text{C}$
$\therefore\text{I}=\frac{2}{9}(8-9\text{x})\sqrt{1+3\text{x}}+\text{C}$

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 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

A factory makes tennis rackets and cricket bats. A tennis racket takes $1.5$ hours of machine time and $3$ hours of craftman's time in its making while a cricket bat takes $3$ hours of machine time and $1$ hour of craftman's time. In a day, the factory has the availability of not more than $42$ hours of machine time and $24$ hours of craftman's time.
  1. What number of rackets and bats must be made if the factory is to work at full capacity?
  2. If the profit on a racket and on a bat is Rs. $20$ and Rs. $10$ respectively, find the maximum profit of the factory when it works at full capacity.
→
Find the shortest distance between the following pairs of lines whose vector equations are:
$\vec{\text{r}}=(1-\text{t})\hat{\text{i}}+(\text{t}-2)\hat{\text{j}}+(3-\text{t})\hat{\text{k}}$ and $\vec{\text{r}}=(\text{s}+1)\hat{\text{i}}+(2\text{s}-1)\hat{\text{j}}-(2\text{s}+1)\hat{\text{k}}$
→
Prove that:
$\begin{vmatrix}\text{a}&\text{b}-\text{c}&\text{c}-\text{b}\\\text{a}-\text{c}&\text{b}&\text{c}-\text{a}\\\text{a}-\text{b}&\text{b}-\text{a}&\text{c}\end{vmatrix}$
$=(\text{a}+\text{b}-\text{c})(\text{b}+\text{c}-\text{a})(\text{c}+\text{a}-\text{b})$
→
Differentiate the following functions with respect to x:
$\text{e}^{\text{ax}}\sec\text{x}\tan2\text{x}$
→
Write the minors and cofactors of element of the first column of the following matrices and hence evaluate the determinant in case:
$\text{A}=\begin{vmatrix}\text{a}&\text{h}&\text{g}\\\text{h}&\text{b}&\text{f}\\\text{g}&\text{f}&\text{c} \end{vmatrix}$
→
Examine the consistency of the system of equation x + y + z = 1; 2x + 3y + 2z = 2; ax + ay + 2az = 4
→
Show that the function $f : R _0 \rightarrow R _0$, defined as $f ( x )=\frac{1}{x}$, is one-one onto, where $R _0$ is the set non-zero real numbers.
Is the result true, if the domain $R _0$ is replaced by N with co-domain being same as $R _0$ ?
→
$R$ is a relation from $\{11, 12, 13\}$ to $\{8, 10, 12\}$ defined by $y = x - 3.$ Then, $R^{-1} $ is:
→
Solve the following differential equation
$\frac{1}{\text{x}}\frac{\text{dy}}{\text{dx}}=\tan^{-1}\text{x},\text{x}\neq0$
→
Verify Rolle's theorem of the following function on the indicated interval
$\text{f}(\text{x})=\log(\text{x}^2+2)-\log3\text{ on }[-1,1]$
→