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" 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

Two cards are drawn from a well shuffled pack of 52 cards. Find the probability distribution of the number of aces.

Find the equation of the plane passing through the origin and perpendicular to each of the planes x + 2y - z = 1 and 3x - 4y + z = 5.

The cartesian equation of a line AB are $\frac{2\text{x}-1}{\sqrt{3}}=\frac{\text{y}+2}{2}=\frac{\text{z}-3}{3}.$ Find the direction cosines of a line parallel to AB.

Find the integrals of the functions in Exercises:
$\tan^4\text{x}$

$\int\frac{1}{1-\sin\frac{\text{x}}{2}}\text{dx}$

Find matrices X and Y, if $2\text{X}-\text{Y}=\begin{bmatrix}6&-6&0\\-4&2&1\end{bmatrix}$ and $\text{X}+2\text{Y}=\begin{bmatrix}3&2&5\\-2&1&-7\end{bmatrix}$

If $\vec{a}, \vec{b}, \vec{c}$ are perpendicular to each other and of equal magnitudes, then prove that the vector $\vec{a}+\vec{b}+\vec{c}$ makes equal angle with $\vec{a}, \vec{b}$ and $\vec{c}$

Evaluate the following definite integrals:
$\int_{0}^\limits{2\pi}\text{e}^{\text{x}}\cos\Big(\frac{\pi}{4}+\frac{\text{x}}{2}\Big)\text{dx}$

Evaluate the following integrals:

$\int\frac{1}{\sqrt{8+3\text{x}-\text{x}^2}}\text{ dx}$

A plane meets the coordinate axes at A, B and C, respectively, such that the centriod of triangle ABC is (1, -2, 3). Find the equation of the plane.