Question
Evaluate: $\int\frac{2}{\text{(1-x)(1+x)}^{2}}\text{dx}$
✓
Answer
$\frac{2}{\text{(1-x)}\text{(1+x}^{2})}=\frac{\text{A}}{\text{1-x}}+\frac{\text{Bx+C}}{\text{1+x}^{2}}$
$2 = A(1 + x^2) + (Bx + C) (1 – x)$
$\Rightarrow 0 = A – B, B – C = 0, A + C = 2$
$\Rightarrow A = B = C = 1$
$\therefore\int\frac{2}{\text{(1-x)(1+x}^{2})}\text{dx}=\int\frac{1}{\text{1-x}}\text{dx}+\frac{\text{x+1}}{\text{x}^{2}+{1}}\text{dx}$
$= – \log |1 – x| + \frac{1}{2} \log (x^2 + 1) + \tan^{-1} x + c.$
$2 = A(1 + x^2) + (Bx + C) (1 – x)$
$\Rightarrow 0 = A – B, B – C = 0, A + C = 2$
$\Rightarrow A = B = C = 1$
$\therefore\int\frac{2}{\text{(1-x)(1+x}^{2})}\text{dx}=\int\frac{1}{\text{1-x}}\text{dx}+\frac{\text{x+1}}{\text{x}^{2}+{1}}\text{dx}$
$= – \log |1 – x| + \frac{1}{2} \log (x^2 + 1) + \tan^{-1} x + c.$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Solve the following differential equation: $(\text{x}^{2} - \text{y}^{2}) \text{dx} + \text{2xy dy =0}$ given that $y = 1$ when $x = 1$
→A farmer mixes two brands P and Q of cattle feed. Brand P, costing Rs. 250 per bag, contains 3 units of nutritional element A, 2.5 units of element B and 2 units of element C. Brand Q costing Rs. 200 per bag contains 1.5 units of nutritional element A, 11.25 units of element B and 3 units of element C. The minimum requirements of nutrients A, B and C are 18 units, 45 units and 24 units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag? What is the minimum cost of the mixture per bag?
An aeroplane can carry a maximum of $200$ passengers. A profit of $Rs. 400$ is made on each first class ticket and a profit of $Rs. 600$ is made on each economy class ticket. The airline reserves at least $20$ seats of first class. However, at least $4$ times as many passengers prefer to travel by economy class to the first class. Determine how many each type of tickets must be sold in order to maximize the profit for the airline. What is the maximum profit.
→Find the mean and variance of the number of tails in three tosses of a coin.
Solve the following differential equation:
$(\text{x}^2-1)\frac{\text{dy}}{\text{dx}}+2(\text{x}+2)\text{y}=2(\text{x}+1)$
→$(\text{x}^2-1)\frac{\text{dy}}{\text{dx}}+2(\text{x}+2)\text{y}=2(\text{x}+1)$
Find the local maximum and local minima, of the function $\text{f(x)} = \sin x - \cos x, 0< x < 2\pi.$ Also find the local maximum and local minimum values.
→The feasible region for a LPP is shown in Evaluate Z = 4x + y at each of the corner points of this region. Find the minimum value of Z, if it exists.
Evaluate the following intregals:
$\int\frac{5\text{x}^2+20\text{x}+6}{\text{x}^2+2\text{x}^2+\text{x}}\ \text{dx}$
→$\int\frac{5\text{x}^2+20\text{x}+6}{\text{x}^2+2\text{x}^2+\text{x}}\ \text{dx}$
Find the equatoion of the plane passing through the points $(2, 2, 1)$ and $(9, 3, 6)$ and perpendicular to the plane $2x + 6y + 6z = 1.$
→A furniture manufacturing company plans to make two products : chairs and tables. From its available resources which consists of 400 square feet to teak wood and 450 man hours. It is known that to make a chair requires 5 square feet of wood and 10 man-hours and yields a profit of Rs. 45, while each table uses 20 square feet of wood and 25 man-hours and yields a profit of Rs. 80. How many items of each product should be produced by the company so that the profit is maximum?