Evaluate : x-5 x-7 d x - Vidyadip

Question

Evaluate : $\int \sqrt{\frac{x-5}{x-7}} \cdot d x$

✓

Answer

$\mathrm{I}=\int \sqrt{\frac{(x-5) \cdot(x-5)}{(x-7) \cdot(x-5)}} \cdot d x=\int \sqrt{\frac{(x-5)^2}{x^2-12 x+35}} \cdot d x$
$
\because \quad \begin{aligned}
x-5 & =A \cdot \frac{d}{d x}\left(x^2-12 x+35\right)+B \\
x-5 & =A(2 x-12)+B \\
& =(2 A) x+(-12 A+B)
\end{aligned}
$
compairing, the co-efficients of like variables and constants
$
\begin{aligned}a
& 2 A=1 \text { and }-12 A+B=-5 \\
\Rightarrow & A=\frac{1}{2} \text { and } B=1
\end{aligned}
$
$\begin{aligned} & \mathrm{I}=\int \frac{\frac{1}{2} \cdot \frac{d}{d x}\left(x^2-12 x+35\right)+(1)}{\sqrt{x^2-12 x+35}} \cdot d x \\ & =\frac{1}{2} \cdot \int \frac{\frac{d}{d x}\left(x^2-12 x+35\right)}{\sqrt{x^2-12 x+35}} \cdot d x+\int \frac{1}{\sqrt{x^2-12 x+35}} \cdot d x \\ & =\mathrm{I}_1+\mathrm{I}_2 \\ & \therefore \quad I_1=\frac{1}{2} \cdot \int \frac{2 x-12}{\sqrt{x^2-12 x+35}} \cdot d x \\ & \text { put } x^2-12 x+35=t \\ & \therefore(2 x-12) \cdot d x=1 \cdot d t \\ & \mathrm{I}_1 \quad=\frac{1}{2} \cdot \int \frac{1}{\sqrt{t}} \cdot d t \\ & =\int \frac{1}{2 \sqrt{t}} \cdot d t \\ & =\sqrt{t}+c_1 \\ & =\sqrt{x^2-12 x+35}+c \\ & \end{aligned}$
$
\begin{aligned}
& \therefore \quad \mathrm{I}_2=\int \frac{1}{\sqrt{x^2-12 x+35}} \cdot d x \\
& =\int \frac{1}{\sqrt{x^2-12 x+36-1}} \cdot d x \\
& =\int \frac{1}{\sqrt{(x-6)^2-(1)^2}} \cdot d x \\
& \because \quad \int \frac{1}{\sqrt{X^2-A^2}} \cdot d x=\log \left(X+\sqrt{X^2-A^2}\right)+c \\
& \mathrm{I}_2 \quad=\log \left((x-6)+\sqrt{(x-6)^2-1}\right)+c_2 \\
& =\log \left((x-6)+\sqrt{x^2-12 x+35}\right)+c_2 \\
&
\end{aligned}
$
Thus, from (i), (ii) and (iii)
$
\begin{aligned}
& \int \sqrt{\frac{x-5}{x-7}} \cdot d x \\
= & \sqrt{x^2-12 x+35}+\log \left((x-6)+\sqrt{x^2-12 x+35}\right)+c \\
& \left(c_1+c_2=c\right)
\end{aligned}
$

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 "Solve the Following Question.(5 Marks)" from Indefinite Integration→Indefinite Integration — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

A class has 15 students whose ages are 14, 17, 15, 14, 21, 17, 19, 20, 16, 18, 20, 17, 16, 19 and 20 years. One student is selected in such a manner that each has the same chance of being chosen and the age X of the student is recorded. What is the probability distribution of the random variable X? Find mean, variance, and standard deviation of X.

Evaluate the following integrals:$\int\frac{\sin2\text{x}}{\sqrt{\sin^4\text{x}+4\sin^2\text{x}-2}}\text{ dx}$

Show that $\text{A}=\begin{bmatrix} -8 & 5 \\ 2 & 4 \end{bmatrix}$ sastisfies the equation $A^2 + 4A - 42I = 0.$ Hence find $A^{-1}.$

$\begin{bmatrix}2&3\\5&7\end{bmatrix}\begin{bmatrix}1&-3\\-2&4\end{bmatrix}=\begin{bmatrix}-4&6\\-9&\text{x}\end{bmatrix}$ find x.

Assume that each born child is equally likely to be a boy or a girl. If a family has two children, then what is the constitutional probability that both are girls? Given that.

The youngest is a girl,
At least one is a girl.

Define a binary operation * on the set {0, 1, 2, 3, 4, 5} as:
$\text{a}\times\text{b}=\begin{cases}\text{a + b},&\text{if }\text{a + b}<6\\\text{a + b}-6,&\text{if }\text{a + b}\geq6\end{cases}$
Show that 0 is the identity for this operation and each element a ≠ 0 of the set is invertible with 6 − a being the inverse of a.

Find the slopes of the tangent and the normal to the following curves at the indicated points:
$\text{y}=(\sin2\text{x}+\cot\text{x}+2)^2\text{at}\text{ x}=\frac{\pi}{2}$

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. If the profit on a racket and on a bat is Rs. 20 and Rs. 10 respectively, find the number of tennis rackets and cricket bats that the factory must manufacture to earn the maximum profit. Make it as an LPP and solve it graphically.

Evaluate the following integrals:
$\int\frac{\text{x}}{(\text{x}-3)\sqrt{\text{x}+1}}\text{ dx}$

If $(\cos\text{x})^{\text{y}}=(\tan\text{y})^{\text{x}},$ Prove that $\frac{\text{dy}}{\text{dx}}=\frac{\log\tan\text{y}-\text{y}\tan\text{x}}{\log\cos\text{x}-\text{x}\sec\text{y cosec y}}$