Integrate the function: 1 9 x^ 2 +6 x+5 - Vidyadip

Question

Integrate the function: $\frac{1}{9 x^{2}+6 x+5}$

✓

Answer

Clearly, $9 x^{2}+6 x+5$ = $(3 x+1)^{2}+(2)^{2}$
$\Rightarrow \int \frac{1}{9 x^{2}+6 x+5} d x=\int \frac{1}{(3 x+1)^{2}+(2)^{2}} d x$
Let 3x + 1 = t
$\Rightarrow$ 3dx = dt
$\therefore~~ \int \frac{1}{(3 x+1)^{2}+(2)^{2}} d x=\frac{1}{3} \int \frac{1}{t^{2}+2^{2}} d t$
$= \frac{1}{3}\left[\frac{1}{2} \tan ^{-1}\left(\frac{t}{2}\right)\right]+C$
$= \frac{1}{6} \tan ^{-1}\left(\frac{3 \mathrm{x}+1}{2}\right)+\mathrm{C}$

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 "4 Marks" from Integrals→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 $\text{y}\sqrt{\text{x}^2+1}=\log\Big(\sqrt{\text{x}^2+1}-\text{x}\Big),$ prove that $\big(\text{x}^2+1\big)\frac{\text{dx}}{\text{dx}}+\text{xy}+1=0$

Find the particular solution of the differential equation $\frac{\text{dy}}{\text{dx}} - \text{3y} \cot \text{x} = \sin \text{2x}, $ given that y = 2 when $\text{x} = \frac{\pi}{2}.$

Evaluate the following intregals:
$\int\frac{\text{x}}{\sqrt{\text{x}^2+6\text{x}+10}}\ \text{dx}$

Show that $\begin{vmatrix}\text{x}-3&\text{x}-4&\text{x}-\alpha\\\text{x}-2&\text{x}-3&\text{x}-\beta\\\text{x}-1&\text{x}-2&\text{x}-\gamma\end{vmatrix}=0,$ where $\alpha,\beta,\gamma$ are in A.P.

Find the value of $\lambda$ for which the lines $\frac{\text{x}-1}{1}=\frac{\text{y}-2}{2}=\frac{\text{z}+3}{\lambda^2}$ and $\frac{\text{x}-3}{1}=\frac{\text{y}-2}{\lambda^2}=\frac{\text{z}-1}{2}$ are coplanar.

Evaluate the following integrals as limit of sum:
$\int\limits^{\frac{\pi}{2}}_{0}\cos\text{x dx}$

Find the equations of the tangent and the normal to the following curves at the indicated points.
$\text{y}=\text{x}^2+4\text{x}+1\text{ at }\text{x}=3$

Find $\frac{\text{dy}}{\text{dx}}$ in the following cases:
(x + y)² = 2axy

Find the cartesian equation of a line passing through (1, -1, 2) and parallel to the line whose equation are $\frac{\text{x}-3}{1}=\frac{\text{y}-1}{2}=\frac{\text{z}+1}{-2}.$ Also, reduce the equation obtained in vector form.

Find the area enclosed by the curve y = -x² and the straight lilne x + y + 2 = 0.