Download App

Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIntegrals4 Marks
Question
Evaluate :
$
\int \frac{x^3}{(x-1)\left(x^2+1\right)} d x
$
✓

Answer

$
\begin{array}{l}
\int \frac{x^3}{(x-1)\left(x^2-1\right)} d x \\
\Rightarrow \int \frac{x^3}{x^3-x^2+x-1} d x
\end{array}
$
Here degree of numerator and denominator is same. So on dividing numerator by denominator :
$
\begin{array}{l}
\int 1+\frac{x^2-x+1}{(x-1)\left(x^2+1\right)} d x \\
\int d x+\int \frac{x^2-x+1}{(x-1)\left(x^2+1\right)} d x
\end{array}
$
Using Partial fraction,
$
\begin{array}{l}
\frac{x^2-x+1}{(x-1)\left(x^2+1\right)}=\frac{A}{(x-1)}+\frac{B x+C}{x^2+1} \\
\Rightarrow x^2-x+1=A\left(x^2+1\right)+(B x+C)(x-1)
\end{array}
$
Putting $x=1 \Rightarrow 1=2 A \quad \therefore A =\frac{1}{2}$
Comparing coefficient of $x^2 \Rightarrow 1= A + B \therefore B =1-\frac{1}{2}$ $=\frac{1}{2}$
Comparing coefficient of $x-1=- B + C \therefore C =-\frac{1}{2}$
$
1+\frac{1}{2(x-1)}+\frac{\frac{1}{2} x-\frac{1}{2}}{x^2+1}
$
$
1+\frac{1}{2(x-1)}+\frac{1}{2} \frac{x}{x^2+1}-\frac{1}{2} \cdot \frac{1}{x^2+1}
$
On integrating,
$
\begin{array}{l}
\int 1 d x+\frac{1}{2} \int \frac{1}{(x-1)}+\frac{1}{2} \int \frac{x}{x^2+1} d x-\frac{1}{2} \int \frac{1}{x^2+1} d x \\
=x+\frac{1}{2} \log |x-1|+\frac{1}{4} \log \left|x^2+1\right|-\frac{1}{2} \tan ^{-1} x+C
\end{array}
$

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

Solve the following systems of linear equations by cramer's rule:
x - 4y - z = 11,
2x - 5y + 2z = 39,
-3x + 2y + z = 1
→
Evaluate: $\int\frac{\sin(\text{x} - \text{a})}{\sin(\text{x + a})}\text{ dx}.$
→
If li, mi, ni ; i = 1, 2, 3 denotes the direction cosines of three mutually perpendicular vectors in space, prove that AAT = I, where
$\text{A}=\begin{bmatrix}\text{l}_1&\text{m}_1&\text{n}_1\\\text{l}_2&\text{m}_2&\text{n}_2\\\text{l}_3&\text{m}_3&\text{n}_3 \end{bmatrix}.$
→
Discuss the continuity of $\text{f(x)}=\sin|\text{x}|$
→
Find the differential equation of system of concentric circles with centre (1, 2).
→
Prove that:
$\begin{vmatrix}\text{a}+\text{b}+2\text{c}&\text{a}&\text{b}\\\text{c}&\text{b}+\text{c}+2\text{a}&\text{b}\\\text{c}&\text{a}&\text{c}+\text{a}+2\text{b} \end{vmatrix}=2(\text{a}+\text{b}+\text{c})^3$
→
Given that $\frac{\text{dy}}{\text{dx}}=\text{e}^\text{-2y}$ and y = 0 when x = 5. Find the value of x when y = 3.
→
Evaluate the following intregals:
$\int\frac{1}{(\text{x}^2+1)(\text{x}^2+2)}\ \text{dx}$
→
Five forces $\overrightarrow{\text{AB}},\ \overrightarrow{\text{AC}},\ \overrightarrow{\text{AD}},\ \overrightarrow{\text{AE}}\text{ and }\overrightarrow{\text{AF}}$ act at the vertex of a regular hexagon ABCDEF. Prove that the resultant is:

$6\ \overrightarrow{\text{AO}}$, where o is the center of hexagon.

→
If A and B are two events such that $2\text{P(A)}=\text{P(B)}=\frac{5}{13}$ and $\text{P}\Big(\frac{\text{A}}{\text{B}}\Big)=\frac{2}{5}$ find $\text{P}(\text{A}\cap\text{B}).$
→