Need Solution for R.D.Sharma Maths Class 12 Chapter 18 Indefinite Integrals Exercise Revision Exercise Question 90 Maths Textbook Solution.

$\frac{\left(4 x^{2}+5 x+6\right)^{\frac{3}{2}}}{6}+\frac{7}{8} \int \frac{x}{2} \sqrt{4 x^{2}+5 x+6}-\frac{1}{8} \ln \left|2 x+\sqrt{4 x^{2}+5 x+6}\right|+c$

Hint:

You must have to know about integration method.

Given:

$\int(2 x+3) \sqrt{4 x^{2}+5 x+6} d x$

Solution:

$\int(2 x+3) \sqrt{4 x^{2}+5 x+6} d x$

$\frac{1}{4} \int(8 x+5+7) \sqrt{4 x^{2}+5 x+6} d x$

$\frac{1}{4} \int(8 x+5) \sqrt{4 x^{2}+5 x+6} d x+7 \int \sqrt{4 x^{2}+5 x+6} d x$

$\text { Let } 4 x^{2}+5 x+6=t$

$\frac{\frac{1}{4}\left(t^{\frac{1}{2}}+1\right)}{\frac{3}{2}}+7 \int \sqrt{\left(2 x+\frac{5}{2}\right)^{2}}+\left(6-\frac{25}{4}\right) d x$

$\frac{1}{4}\left(\frac{2\left(\sqrt{4 x^{2}+5 x+6}\right)}{3} \cdot\left(4 x^{2}+5 x+6\right)+7 \int \sqrt{\left(2 x+\frac{5}{2}\right)^{2}}+\frac{1}{4} d x\right.$

$\int \frac{\left(4 x^{2}+5 x+6\right)^{\frac{3}{2}}}{6}+\frac{7}{4} \sqrt{\left(2 x+\frac{5}{2}\right)^{2}}+\left(\frac{1}{2}\right)^{2} d x$

$\int \frac{\left(4 x^{2}+5 x+6\right)^{\frac{3}{2}}}{6}+\frac{7}{4} \sqrt{\left(2 x+\frac{5}{2}\right)^{2}}+\left(\frac{1}{2}\right)^{2} d x$

$\inline \frac{\left(4 x^{2}+5 x+6\right)^{\frac{3}{2}}}{6}+\frac{7}{4} \sqrt{\left(2 x+\frac{5}{2}\right)^{2}\left(-\frac{1}{2}\right)^{2}}-\frac{1}{4} \times \frac{1}{2} \ln \left|2 x+\sqrt{4 x^{2}+5 x+6}\right|+c$

$\frac{\left(4 x^{2}+5 x+6\right)^{\frac{3}{2}}}{6}+\frac{7}{8} \int \frac{x}{2} \sqrt{4 x^{2}+5 x+6}-\frac{1}{8} \ln \left|2 x+\sqrt{4 x^{2}+5 x+6}\right|+c$