Evaluate the following integrals: 1 x^2-10x+34 dx - Vidyadip

Question

Evaluate the following integrals:
$\int\frac{1}{\text{x}^2-10\text{x}+34}\text{dx}$

✓

Answer

Let $\text{I}=\int\frac{1}{\text{x}^2-10\text{x}+34}\text{dx}$
$=\int\frac{1}{\text{x}^2-2\text{x}\times5+(5)^2-(5)^2+34}\text{dx}$
$=\int\frac{1}{(\text{x}-5)^2+9}\text{dx}$
Let $(\text{x}-1)=\text{t} \dots(1)$
$\Rightarrow\text{dx = dt}$
so,
$\text{I}=\int\frac{1}{\text{t}^2+(3)^2}\text{dt}$
$\text{I}=\frac{1}{3}\tan^{-1}\big(\frac{\text{t}}{3}\big)+\text{C}$ $\Big[\text{since,}\int\frac{1}{\text{x}^2+\text{a}^2}\text{dx}=\frac{1}{\text{a}}\tan^{-1}\big(\frac{\text{x}}{2}\big)+\text{C}\Big]$
$\text{I}=\frac{1}{3}\tan^{-1}\Big(\frac{\text{x}-5}{3}\Big)+\text{C}$ [using (1)]

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

Evaluate the following integrals:
$\int^\limits1_{-1}|2\text{x}+1|\text{dx}$

Prove that the diagonals of a rhombus are perpendicular bisectors of each other.

Evaluate the following integrals:
$\int_{0}^\limits{\frac{\pi}{2}}\frac{\sin\text{x}\cos\text{x}}{1+\sin^4\text{x}}\text{ dx}$

Integrate the function: $\frac{1}{{1 - \tan x}}$

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{\text{x}}+\sin\Big(\frac{\text{y}}{\text{x}}\Big)$

If a unit vector $\vec a$ makes angles $\frac{\pi}{3}$ with $\hat i$, $\frac{\pi}{4}$ with $\hat j$ and an acute angle $\theta$ with $\hat k$, then find $\theta$ and hence, the components of $\vec a $.

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 craftsman'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 crickets bats that the factory must manufacture to earn the maximum profit.Make it as an L.P.P. and solve graphically.

A factory manufactures two types of screws A and B, each type requiring the use of two machines, an automatic and a hand-operated. It takes 4 minutes on the automatic and 6 minutes on the hand-operated machines to manufacture a packet of screws ‘A’ while it takes 6 minutes on the automatic and 3 minutes on the hand-operated machine to manufacture a packet of screws ‘B’. Each machine is available for at most 4 hours on any day. The manufacturer can sell a packet of screws ‘A’ at a profit of 70 paise and screws ‘B’ at a profit of Rs. 1. Assuming that he can sell all the screws he manufactures, how many packets of each type should the factory owner produce in a day in order to maximize his profit? Formulate the above LPP and solve it graphically and find the maximum profit.

Find a point on the curve $y = x^3 + 1$ where the tangent is parallel to the chord joining (1, 2) and (3, 28).

Evaluate the following:
$\tan\frac{1}{2}\Big(\cos^{-1}\frac{\sqrt5}{3}\Big)$