Evaluate the following definite integrals: _ -1 ^11 x^2+2x+5 dx

Question

Evaluate the following definite integrals:
$\int_{-1}^\limits{1}\frac{1}{\text{x}^2+2\text{x}+5}\text{ dx}$

✓

Answer

Let $\text{I}=\int_{-1}^\limits{1}\frac{1}{\text{x}^2+2\text{x}+5}\text{ dx}$ Then,
$\text{I}=\int_{-1}^\limits{1}\frac{1}{\big(\text{x}^2+2\text{x}+1\big)+4}\text{ dx}$
$\Rightarrow\text{I}=\int_{-1}^\limits{1}\frac{1}{(\text{x}+1)^2+2^2}\text{ dx}$
$\Rightarrow\text{I}=\frac{1}{2}\Big[\tan^{-1}\frac{(\text{x}+1)}{2}\Big]^1_{-1}$
$\Rightarrow\text{I}=\frac{1}{2}\Big(\frac{\pi}{4}\Big)$
$\Rightarrow\text{I}=\frac{\pi}{8}$

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 "3 Marks Question" 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

Let $\overrightarrow{ a }=\hat{ i }+4 \hat{ j }+2 \hat{ k }, \overrightarrow{ b }=3 \hat{ i }-2 \hat{ j }+7 \hat{ k }$ and $\overrightarrow{ c }= 2 \hat{ i }-\hat{ j }+4 \hat{ k }$.Find a vector $\vec{d}$ such that $i t$ is perpendicular to both $\vec{a}$ and $\vec{b}$ and $\vec{c} \cdot \vec{d}=27$.

→

For each of the differential equations in find the general solution:
$\frac{\text{dy}}{\text{dx}}=\frac{1-\text{cos} \ \text{x}}{1+\text{cos} \ \text{x}}$

→

Is the binary operation $\ast,$ defined on set N, given by a$a^{\ast}b = \frac{a + b}{2}$ for all $a, b\varepsilon N,$ commutative?
Is the above binary operation $\ast$ associative?

→

Find the angle between the lines $2\text{x}=3\text{y}=-\text{z}$ and $6\text{x}=-\text{y}=-4\text{z}.$

→

Minimise $Z = 3x + 5y$ subject to the constraints:
$x+2 y \geqslant 10$
$x+y \geqslant 6$
$3 x+y \geqslant 8$
$x, y \geqslant 0$

→

If the mean and variance of a binomial distribution are 4 and 3, respectively, find the probability of no success.

→

Evaluate the following integrals:$\int\frac{\cos2\text{x}}{\sqrt{\sin^22\text{x}+8}}\text{ dx}$

→

If $\text{y}=\sqrt{\tan\text{x}+\sqrt{\tan\text{x}+\sqrt{\tan\text{x}+\ .... \text{to }\infty}}}$ prove that $\frac{\text{dy}}{\text{dx}}=\frac{\sec^2\text{x}}{2\text{y}-1}$

→

Evalute the following integrals:
$\int\frac{10\text{x}^9+10^\text{x}\log_\text{e}10}{10^\text{x}+\text{x}^{10}}\text{dx}$