Find the integral: d x 3 x^ 2 +13 x-10 - Vidyadip

Question

Find the integral: $\int \frac{d x}{3 x^{2}+13 x-10}$

✓

Answer

We write the denominator of the integrand,
$3x^2 + 13x - 10 = 3\left(x^{2}+\frac{13 x}{3}-\frac{10}{3}\right)$
= $3\left[\left(x+\frac{13}{6}\right)^{2}-\left(\frac{17}{6}\right)^{2}\right]$ (completing the square)
Thus, $\int \frac{d x}{3 x^{2}+13 x-10}=\frac{1}{3} \int \frac{d x}{\left(x+\frac{13}{6}\right)^{2}-\left(\frac{17}{6}\right)^{2}}$
Put $x+\frac{13}{6}=t$. Then dx = dt
Therefore, $\int \frac{d x}{3 x^{2}+13 x-10}=\frac{1}{3} \int \frac{d t}{t^{2}-\left(\frac{17}{6}\right)^{2}}$
= $\frac{1}{3 \times 2 \times \frac{17}{6}} \log \left|\frac{t-\frac{17}{6}}{t+\frac{17}{6}}\right|+\mathrm{C}_{1}$
= $\frac{1}{17} \log \left|\frac{x+\frac{13}{6}-\frac{17}{6}}{x+\frac{13}{6}+\frac{17}{6}}\right|+C_{1}$
= $\frac{1}{17} \log \left|\frac{6 x-4}{6 x+30}\right|+C_{1}$
= $\frac{1}{17} \log \left|\frac{3 x-2}{x+5}\right|+C_{1}+\frac{1}{17} \log \frac{1}{3}$
= $\frac{1}{17} \log \left|\frac{3 x-2}{x+5}\right|+C$ where, $\mathrm{C}=\mathrm{C}_{1}+\frac{1}{17} \log \frac{1}{3}$

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 "1 Marks Question" 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

Consider a binary operation * on the set {1, 2, 3, 4, 5} given by the following multiplication table (Table 1.2).
Is * commutative?
(Hint: use the following table)

*	1	2	3	4	5
1	1	1	1	1	1
2	1	2	1	2	1
3	1	1	3	1	1
4	1	2	1	4	1
5	1	1	1	1	5

Write the differrntial equation representing famliy of curve y = mx, where m is arbitrary constant.

The rate of change of the area of a circle with respect to its radius r at $r = 6$ cm is

$\text{Evaluate} : \sin \Bigg[\frac{\pi}{3} - \sin^{-1} \bigg( - \frac{1}{2}\bigg)\Bigg]$

Evaluate the following:
$\tan^{-1}\Big(\tan\frac{\pi}{3}\Big)$

If $(2\ \ 1\ \ 3) \begin{pmatrix} -1 & 0 & -1 \\ -1 & 1 & 0 \\ 0 & 1 & 1 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix} = \text{A},$ then write the order of matrix A.

Find $\int \frac{x^{4} d x}{(x-1)\left(x^{2}+1\right)}$

Let $A = \left[\begin{array}{ccc} {1} & {\sin \theta} & {1} \\ {-\sin \theta} & {1} & {\sin \theta} \\ {-1} & {-\sin \theta} & {1} \end{array}\right]$ where $0 \leq \theta \leq 2 \pi$. Then

Find the rate of change of the area of a circular disc with respect to its circumference when the radius is 3cm.

Using integration, find area of the bounded between the line $x = 2$ and the parabola $y^2 = 8x$.