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

Question

Evalute the following integrals:
$\int\frac{\cos2\text{x}+\text{x}+1}{\text{x}^2+\sin2\text{x}+2\text{x}}\text{dx}$

✓

Answer

Let $\text{I}=\int\frac{\cos2\text{x}+\text{x}+1}{\text{x}^2+\sin2\text{x}+2\text{x}}\text{dx}$
Putting $\text{x}^2+\sin2\text{x}+2\text{x}=\text{t}$
$\Rightarrow2\text{x}+2\cos2\text{x}+2=\frac{\text{dt}}{\text{dx}}$
$\Rightarrow(\text{x}+\cos2\text{x}+1)\text{dx}=\frac{\text{dt}}{2}$
$\therefore\text{I}=\frac{1}{2}\int\frac{1}{\text{t}}\text{dt}$
$=\frac{1}{2}\text{ln}|\text{t}|+\text{C}$
$=\frac{1}{2}\text{ln}|\text{x}^2+\sin2\text{x}+2\text{x}|+\text{C}$
$\big[\because\text{t}=\text{x}^2+\sin2\text{x}+2\text{x}\big]$

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 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

If $\big|\vec{\text{a}}\times\vec{\text{b}}\big|^2+\big|\vec{\text{a}}.\vec{\text{b}}\big|^2=400$ and $|\vec{\text{a}}|=5,$ then write the value of $\big|\vec{\text{b}}\big|.$

Find the mean and standard deviation of the following probability distributions:

$x_i$	-2	-1	0	1	2
$p_i$	0.1	0.2	0.4	0.2	0.1

Write a value of $\int\frac{\text{a}^{\text{x}}}{3+\text{a}^{\text{x}}}\text{ dx}$

Evaluate the following integrals:
$\int\frac{1}{\text{x}^2+6\text{x}+13}\text{dx}$

Find the real solutions of the equation $\tan^{-1}\sqrt{\text{x}(\text{x}+1)}+\sin^{-1}\sqrt{\text{x}^2+\text{x}+1}=\frac{\pi}{2}.$

Solve the following differential equations:
$\frac{\text{dy}}{\text{dx}}=2\text{xy, y}(0)=1$

Check the commutativity and associativity of the following binary operations:
'*' on Q defined by $a * b = ab^2$ for all $a, b \in Q.$

There are two boxes, namely box$-I$ and box$-II.$ Box$-I$ contains $3$ red and $6$ black balls. Box$-II$ contains $5$ red and $5$ black balls. One of the two boxes, is selected at random and a ball is drawn at random. The ball drawn is found to be red. Find the probability that this red ball comes out from box$-II.$

Are the following set of ordered pairs functions? If so, examine whether the mapping is injective or surjective:
{(x, y): x is a person, y is the mother of x}

Find the sine of the angle between the vectors $\vec{\text{a}}=3\hat{\text{i}}+\hat{\text{j}}+2\hat{\text{k}}$ and $\vec{\text{b}}=2\hat{\text{i}}-2\hat{\text{j}}+4\hat{\text{k}}.$