Evalute the following integrals: 1 1+x+x^2+x^3 dx - Vidyadip

Question

Evalute the following integrals:
$\int\frac{1}{1+\text{x}+\text{x}^2+\text{x}^3}\text{ dx}$

✓

Answer

We have,
$\text{I}=\int\frac{\text{dx}}{1+\text{x}+\text{x}^2+\text{x}^3}$
$=\int\frac{\text{dx}}{(\text{x}+1)(\text{x}^2+1)}$
$=\int\frac{\text{dx}}{(\text{x}+1)(\text{x}^2+1)}$
Let $\frac{1}{(\text{x}+1)(\text{x}^2+1)}=\frac{\text{A}}{\text{x}+1}+\frac{\text{Bx}+\text{C}}{\text{x}^2+1}$
$\Rightarrow\frac{1}{(\text{x}+1)(\text{x}^2+1)}=\frac{\text{A}(\text{x}^2+1)+(\text{Bx}+\text{C})(\text{x}+1)}{(\text{x}+1)(\text{x}^2+1)}$
$\Rightarrow1=\text{A}(\text{x}^2+1)+\text{Bx}^2+\text{Bx}+\text{Cx}+\text{C}$
$\Rightarrow1=(\text{A}+\text{B})\text{x}^2+(\text{B}+\text{C})\text{x}+(\text{A}+\text{C})$
Equating coefficient of like terms
A + B = 0 ...(1)
B + C = 0 ...(2)
A + C = 1 ...(3)
Solving (1), (2) and (3), we get
$\text{A}=\frac{1}{2}$
$\text{B}=-\frac{1}{2}$
$\text{C}=\frac{1}{2}$
$\therefore\text{I}=\frac{1}{2}\int\frac{\text{dx}}{\text{x}+1}+\frac{1}{2}\int\Big(\frac{-\text{x}+1}{\text{x}^2+1}\Big)\text{dx}$
$=\frac{1}{2}\int\frac{\text{dx}}{\text{x}+1}-\frac{1}{2}\int\frac{\text{x dx}}{\text{x}^2+1}+\frac{1}{2}\int\frac{\text{dx}}{\text{x}^2+1^2}$
Let $\text{x}^2+1=\text{dt}$
$\Rightarrow2\text{x dx}=\text{dt}$
$\Rightarrow\text{x dx}=\frac{\text{dt}}{2}$
$\therefore\text{I}=\frac{1}{2}\int\frac{\text{dx}}{\text{x}+1}-\frac{1}{4}\int\frac{\text{dt}}{\text{t}}+\frac{1}{2}\int\frac{\text{dx}}{\text{x}^2+1^2}$
$=\frac{1}{2}\log|\text{x}+1|-\frac{1}{4}\log|\text{t}|+\frac{1}{2}\tan^{-1}(\text{x})+\text{C}$
$=\frac{1}{2}\log|\text{x}+1|-\frac{1}{4}\log|\text{x}^2+1|+\frac{1}{2}\tan^{-1}(\text{x})+\text{C}$

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 "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

$\text{Find:}\int \frac{\text{x}\ \text{dx}}{(2 + \text{x}^{\text{2}}) (4 + \text{e}^{4\text{}})}$

Find the points of discontinuity, if any of the following function:
$\text{f(x)}=\begin{cases}\frac{\text{x}^4+\text{x}^3+2\text{x}^2}{\tan^{-1}\text{x}},&\text{if }\text{ x}\neq0\\10,&\text{if }\text{ x}=0\end{cases}$

If $\text{xy}\log(\text{x}+\text{y})=1,$ prove that $\frac{\text{dx}}{\text{dx}}=-\frac{\text{y}(\text{x}^2\text{y}+\text{x}+\text{y})}{\text{x}(\text{xy}^2+\text{x}+\text{y})}$

Evaluate the follwing intregals:
$\int\frac{1}{\text{x}^4-1}\text{ dx}$

Solve the following differential equation:
$\text{dx + xdy}=\text{e}^{-\text{y}}\sec^2\text{y dy}$

Find the condition for the following set of curves to intersect orthogonally
$\frac{\text{x}^2}{\text{a}^2}-\frac{\text{y}^2}{\text{b}^2}=1\text{ and }\frac{\text{x}^2}{\text{A}^2}-\frac{\text{y}^2}{\text{B}^2}=1$

A random variable $X$ has the following probability distribution:

$X$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$
$P(X)$	$0$	$K$	$2K$	$2K$	$3K$	$K^2$	$2K^2$	$7K^2+ K$

$K.$
$P(X < 3).$
$P(X > 6).$
$P(0 < X < 3).$

Differentiate the following functions with respect to x:
$\tan^{-1}\Big(\frac{\text{a}+\text{bx}}{\text{b}-\text{ax}}\Big)$

Show that the function f defined as follows,
$\text{f(x)}=\begin{cases}3\text{x}-2, & 0<\text{x}\leq1\\2\text{x}^2-\text{x,} & 1<\text{x}\leq2\\5\text{x}-4,&\text{x}>2\end{cases}$
is countinuous at x = 2, but not differentiable there at x = 2.

There are three urns A, B, and C. Urn A contains 4 red balls and 3 black balls. urn B contains 5 red balls and 4 black balls. Urn C contains 4 red and 4 black balls. One ball is drawn from each of these urns. What is the probability that 3 balls drawn consists of 2 red balls and a black ball?