Evaluate the following integrals: 1 3x^2+5x+7 dx - Vidyadip

Question

Evaluate the following integrals:$\int\frac{1}{\sqrt{3\text{x}^2+5\text{x}+7}}\text{ dx}$

✓

Answer

$\int\frac{\text{dx}}{\sqrt{3\text{x}^2+5\text{x}+7}}$
$=\int\frac{\text{dx}}{\sqrt{3\big(\text{x}^2+\frac{5}{3}\text{x}+\frac{7}{3}}\big)}$
$=\frac{1}{\sqrt3}\int\frac{\text{dx}}{\sqrt{\text{x}^2+\frac{5}{3}\text{x}+\big(\frac{5}{6}\big)^2-\big(\frac{5}{6}\big)^2+\frac{7}{3}}}$
$=\frac{1}{\sqrt3}\int\frac{\text{dx}}{\sqrt{\big(\text{x}+\frac{5}{6}\big)^2-\frac{25}{36}+\frac{7}{3}}}$
$=\frac{1}{\sqrt3}\int\frac{\text{dx}}{\sqrt{\big(\text{x}+\frac{5}{6}\big)^2+\frac{-25+84}{36}}}$
$=\frac{1}{\sqrt3}\int\frac{\text{dx}}{\sqrt{\big(\text{x}+\frac{5}{6}\big)^2+\frac{59}{36}}}$
$=\frac{1}{\sqrt3}\int\frac{\text{dx}}{\sqrt{\big(\text{x}+\frac{5}{6}\big)^2+\Big(\sqrt{\frac{59}{36}}\Big)^2}}$
$=\frac{1}{\sqrt3}\log\Bigg|\text{x}+\frac{5}{6}+\sqrt{\big(\text{x}+\frac{5}{6}\big)^2+\frac{59}{36}}\Bigg|+\text{C}$
$=\frac{1}{\sqrt3}\log\Bigg|\text{x}+\frac{5}{6}+\sqrt{\text{x}^2+\frac{5}{3}\text{x}+\frac{7}{3}}\Bigg|+\text{C}$

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 "Solve the Following Question.(5 Marks)" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Differentiate $\sin^{-1}\Big(\frac{2\text{x}}{1+\text{x}^2}\Big)$ with respect to $\tan^{-1}\Big(\frac{2\text{x}}{1-\text{x}^2}\Big),$ if -1 < x < 1.

Let f be a real function given by $\text{f(x)}=\sqrt{\text{x}-2}.$ Find the following:
fofof
Also, show that fof ≠ $f^2.$

If $\text{y}=\text{x}\sin(\text{a}+\text{y}),$ prove that $\frac{\text{dy}}{\text{dx}}=\frac{\sin^2(\text{a}+\text{y})}{\sin(\text{a}+\text{y})-\text{y}\cos(\text{a}+\text{y})}$

Find the percentage error in calculating the surface area of a cubical box if an error of 1% is made in measuring the lengths of edges of the cube.

For the matrix $\text{A}=\begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & -3 \\ 2 & -1 & 3 \end{bmatrix}.$ Show that $A^{-3} - 6A^2 + 5A + 11I_3 = 0$ Hence, find $A^{-1}.$

Evaluate the following intregals:
$\int\frac{18}{(\text{x}+2)(\text{x}^2+4)}\text{ dx}$

If $\vec{\text{p}}$ and $\vec{\text{q}}$ are unit vectors forming an angle of 30°; find the area of the parallelogram having $\vec{\text{a}}=\vec{\text{p}}+2\vec{\text{q}}$ and $\vec{\text{b}}=2\vec{\text{p}}+\vec{\text{q}}$ as its diagonals.

A and B are two independent events. The probability that A and B occur is $\frac{1}{6}$ and the probability that neither of them occurs is $\frac{1}{3}$. Find the probability of occurrence of two events.

Solve the following determinant equations:
$\begin{vmatrix}\text{x}+1&3&5\\2&\text{x}+2&5\\2&3&\text{x}+4\end{vmatrix}=0$

Differentiate $\tan ^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)$ w.r.t. $\cos ^{-1}\left(\sqrt{\frac{1+\sqrt{1+x^2}}{2 \sqrt{1+x^2}}}\right)$