Integrate the function in Exercise: 1 8+3x-x^2 - Vidyadip

Question

Integrate the function in Exercise:
$\frac{1}{\sqrt{8+3\text{x}-\text{x}^2}}$

✓

Answer

$\int\frac{1}{\sqrt{8+3\text{x}-\text{x}^2}}\text{ dx}$
$=\int\frac{1}{\sqrt{-\text{x}^2+3\text{x}+8}}\text{ dx}$
$=\int\frac{1}{\sqrt{-\big(\text{x}^2-3\text{x}-8\big)}}\text{ dx}$
$=\int\frac{1}{\sqrt{-\bigg\{\text{x}^2-3\text{x}+\bigg(\frac{3}{2}\bigg)^2-\bigg(\frac{3}{2}\bigg)^2-8\bigg\}}}\text{ dx}$
$=\int\frac{1}{\sqrt{-\bigg\{\bigg(\text{x}-\frac{3}{2}\bigg)^2-\frac{41}{4}\bigg\}}}\text{ dx}$
$=\int\frac{1}{\sqrt{\bigg(\frac{\sqrt{41}}{2}\bigg)^2-\bigg(\text{x}-\frac{3}{2}\bigg)^2}}\text{ dx}$
$=\sin^{-1}\frac{\text{x}-\frac{3}{2}}{\frac{\sqrt{41}}{2}}+\text{c}=\sin^{-1}\bigg(\frac{2\text{x}-3}{\sqrt{41}}\bigg)+\text{c}$

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 "3 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

Let $\vec{\text{a}}=\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}},\vec{\text{b}}=\hat{\text{i}}$ and $\vec{\text{c}}=\text{c}_1\hat{\text{i}}+\text{c}_2\hat{\text{j}}+\text{c}_3\hat{\text{k}}.$ Then,
If C₂ = -1 and C₃ = 1, show that no value of C₁ can make $\vec{\text{a}},\vec{\text{b}}$ and $\vec{\text{c}}$ coplanar.

If $\text{x}=3\sin\text{t}-\sin3\text{t},\text{y}=3\cos3\text{t}-\cos3\text{t}$ find $\frac{\text{dy}}{\text{dx}}\text{ at t}=\frac{\pi}{3}$

Give examples of two functions f : N → N and g : N → N, such that gof is onto but f is not onto.

Using determinants, find the value of k so that the points (k, 2 - 2k), (-k + 1, 2k) and (-4 - k, 6 - 2k) may be collinear.

Find the vector equation of the line passing through the point (1, 2, -4) and perpendicular to the two lines:$ \frac { x - 8 } { 3 } = \frac { y + 19 } { - 16 } = \frac { z - 10 } { 7 }$ and $ \frac { x - 15 } { 3 } = \frac { y - 29 } { 8 } = \frac { z - 5 } { - 5 }.$

Find the intervals in which the following functions are increasing or decreasing.
f(x) = 2x³ - 9x² - 12x + 1

Form the differential equation corresponding to $\text{y}^2=\text{a}(\text{b}-\text{x}^2)$ bt eliminating a and b.

A coin is tossed three times, if head occurs on first two tosses, find the probability of getting head on third toss.

Evaluate the following integrals:

$\int\frac{\log(\log\text{x})}{\text{x}}\text{dx}$

$A$ and $B$ are two points. The position vector of point $A$ is $6 \vec{b}-2 \vec{a}$. A point $P$ divides line $A B$ in the ratio of $1: 2$. If the position vector of $P$ is $\vec{a}-\vec{b}$, then find the position vector of $B$.