Download App

Definite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDefinite Integrals5 Marks
Question
Evaluate the following integrals:
$\int\limits^1_0\frac{1-\text{x}^2}{\text{x}^4+\text{x}^2+1}\text{ dx}$
✓

Answer

Let $\text{I}=\int^\limits1_0\frac{1-\text{x}^2}{\text{x}^4+\text{x}^2+1}\text{ dx}$
$=-\int\frac{\text{x}^2-1}{\text{x}^4+\text{x}^2+1}\text{ dx}$
$=-\int\frac{1-\frac{1}{\text{x}^2}}{\text{x}^2+1+\frac{1}{\text{x}^2}}\text{ dx}$
$=-\int\frac{1-\frac{1}{\text{x}^2}}{\Big(\text{x}+\frac{1}{\text{x}}\Big)^2-1}\text{ dx}$
Let, $\text{x}+\frac{1}{\text{x}}=\text{t}$
$\Big(1-\frac{1}{\text{x}^2}\Big)\text{dx}=\text{dt}$
Then integral becames,
$\text{I}=-\int\frac{1}{\text{t}^2-1}\text{dt}$
$=-\frac{1}{2}\log\Big|\frac{\text{t}-1}{\text{t}+1}\Big|$
$=\frac{1}{2}\log\Big|\frac{\text{t}+1}{\text{t}-1}\Big|$
$=\frac{1}{2}\log\Big|\frac{\text{x}^2+\text{x}+1}{\text{x}^2-\text{x}+1}\Big|$
i,e., $\int\frac{1-\text{t}^2}{\text{x}^4+\text{x}^2+1}\text{ dx}=\frac{1}{2}\log\Big|\frac{\text{x}^2+\text{x}+1}{\text{x}^2-\text{x}+1}\Big|$
$\int^\limits1_0\frac{1-\text{x}^2}{\text{x}^4+\text{x}^2+1}\text{ dx}=\Big[\frac{1}{2}\log\Big|\frac{\text{x}^2+\text{x}+1}{\text{x}^2-\text{x}+1}\Big|\Big]^1_0$
$=\frac{1}{2}\log3$

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 Definite Integrals→Definite 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 intregals:
$\int\frac{1}{2+\sin\text{x}+\cos\text{x}}\text{dx}$
→
If A and B are two independent events such that $\text{P}\overline{\text({A}} \cap \text{B}) = \frac{2}{15} \text{and P} (\text{A}\cap \overline{\text{B)}} = \frac{1}{6},$ then find P(A) and P(B).
→
Evaluate the following definite integrals:
$\int\limits_{0}^{\pi}\text{e}^{2\text{x}}\sin\Big(\frac{\pi}{4}+{\text{x}}\Big)\text{dx}$
→
Find the volume of the largest cylinder that can be inscribed in a sphere of radius r.
→
A bank pays interest by continuous compounding, that is, by treating the interest rate as the instantaneous rate of change of principal. Suppose in an account interest accrues at $8\%$ per year, compounded continuously. Calculate the percentage increase in such an account over one year.
→
Solve the following systems of linear equations by cramer's rule:
$6x + y - 3z = 5,$
$x + 3y - 2z = 5,$
$2x + y + 4z = 8$
→
A random variable X takes the values 0, 1, 2 and 3 such that:
P(X = 0) = P(X > 0) = P(X < 0); P(X = -3) = P(X = -2) = P(X = -1); P(X = 1) = P(X = 2) = P(X = 3).
Obtain the probability distribution of X.
→
Solve the differential equation $(\text{y}+3\text{x}^2)\frac{\text{dx}}{\text{dy}}=\text{x}$
→
Find the inverse of the following matrices by using elementry row transformation:$\begin{bmatrix} 1 & 2 & 0 \\ 2 & 3 & -1 \\ 1 & -1 & 3 \end{bmatrix}$
→
Show that the maximum volume of the cylinder which can be inscribed in a sphere of radius $5\sqrt{3}\text{cm}$ is $500\pi\text{cm}^{3}$ .
→