Evaluate: 2 (1-x)(1+x)^ 2 dx - Vidyadip

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 + x²) + (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 (x² + 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.

Start Generating Free

Explore more

More "4 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

If $\text{A}=\begin{bmatrix} 1 & -2 & 3 \\ 0 & -1 & 4 \\ -2 & 2 & 1 \end{bmatrix},$ find (A^T)^-1.

Maximize Z = 3x₁ + 4x₂, if possible,

Subject to the constraints

$\text{x}_1-\text{x}_2\leq-1$

$-\text{x}_1+\text{x}_2\leq0$

$\text{x}_1,\text{x}_2\geq0$

Find the vector and cartesian equations of the line passing through the point (2, 1, 3) and perpendicular to the lines$\frac{\text{x}-1}{1}=\frac{\text{y}-2}{2}=\frac{\text{z}-3}{3}$ and$\frac{\text{x}}{-3}=\frac{\text{y}}{2}=\frac{\text{z}}{5}.$

$\text{If f(x)} = \begin{vmatrix} a & -1 & 0 \\ ax & a & -1 \\ ax^{2} & ax & a \end{vmatrix} , $ using properties of determinants find the value of $\text{f}(2x) - \text{f}(x).$

Find the particular solution of the differential equation
x (x² – 1) $\frac{\text{dy}}{\text{dx}}$ = 1; y = 0 when x = 2.

A total amount of ₹ 7000 is deposited in three different saving bank accounts with annual interest rates 5%, 8% and 8$\frac{1}{2}$% respectively. The total annual interest from these three accounts is ₹ 550. Equal amounts have been deposited in the 5% and 8% saving accounts. Find the amount deposited in each of the three accounts, with the help of matrices.

Find the area enclosed by the parabola y = 5x² and y = 2x² + 9.

Find the vector equation of the plane passing through three point with position vectors $\hat{\text{i}}+\hat{\text{j}}-2\hat{\text{k}},2\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}$ and $\hat{\text{i}}+2\hat{\text{j}}+\hat{\text{k}}.$ Also, find coordinates of the point of intersection of this plane and the line $\vec{\text{r}}=3\hat{\text{i}}-\hat{\text{j}}-\hat{\text{k}}+\lambda(2\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}}).$

Evaluate the following integrals:
$\int\frac{\text{e}^{\text{x}}}{\text{x}}\Big\{\text{x}(\log\text{x})^2+2\log\text{x}\Big\}\text{dx}$

If P is a point and ABCD is a quadrilateral and $\overrightarrow{\text{AP}}+\overrightarrow{\text{PB}}+\overrightarrow{\text{PD}}=\overrightarrow{\text{PC}}$, show that ABCD is a parallelogram.