Find: (3x + 1) 4 - 3x - 2x^ 2 dx - Vidyadip

Question

Find: $\int(3x + 1)\sqrt{4 - 3x - 2x^{2 } }\text{ dx}$

✓

Answer

$\int(3\text{x} + 1)\sqrt{4 - 3\text{x} - 2\text{x}^{2}}\text{ dx} = -\frac{3}{4}\int(-4\text{x} - 3)\sqrt{4 - \text{3x} - \text{2x}^{2}} \text{ dx} - \frac{5}{4}\int\sqrt{4 - 3\text{x} - 2\text{x}^{2}} \text{ dx}$
$= -\frac{1}{2}(4 - \text{3x} - \text{2x}^{2})^{3/2}-\frac{5}{4}\sqrt{2}\int\sqrt{\bigg(\frac{\sqrt{41}}{4}\bigg)^{2} - \bigg(\text{x} + \frac{3}{4}\bigg)^{2}} \text{dx}$
$= -\frac{1}{2}(4 - \text{3x} - \text{2x}^{2})^{3/2}-\frac{5}{4}\sqrt{2} \left\{\frac{4\text{x} + 3}{8} \sqrt{\frac{41}{16}- \bigg(\text{x} + \frac{3}{4}\bigg)^{2}} + \frac{41}{32}.\sin^{-1}\bigg(\frac{\text{4x + 3}}{\sqrt{41}}\bigg)\right\} + \text{C}$
$= -\frac{1}{2}(4 - \text{3x} - \text{2x}^{2})^{3/2}-\frac{5}{4}\left\{\frac{\text{4x + 3}}{8} \sqrt{4 - \text{3x - 2x}^{2}} + \frac{41\sqrt{2}}{32}.\sin^{-1}\bigg(\frac{\text{4x + 3}}{\sqrt{41}}\bigg)\right\} + \text{C}$

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 "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 $\text{x}=\text{a}(\cos\theta+\theta\sin\theta),\text{y}=\text{a}(\sin\theta-\theta\cos\theta)$ prove that $\frac{\text{d}^2\text{x}}{\text{d}\theta^2}=\text{a}(\cos\theta-\theta\sin\theta),\frac{\text{d}^2}{\text{d}\theta^2}$ $=\text{a}(\sin\theta-\theta\cos\theta)\ \text{and}\ \frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{\sec^3\theta}{\text{a}\theta}$

Using integration, find the area of the region bounded by the triangle whose vertices are (–1, 2), (1, 5) and (3, 4).

Evaluate: $\int\limits^\frac{3}{2}_{0}|x \cos\pi x| dx$

Evaluate the following integrals:
$\int\tan^5\text{x}\text{ dx}$

Prove that:
$\begin{vmatrix}\text{z}&\text{x}&\text{y}\\\text{z}^2&\text{x}^2&\text{y}^2\\\text{z}^4&\text{x}^4&\text{y}^4 \end{vmatrix}=\begin{vmatrix}\text{x}&\text{y}&\text{z}\\\text{x}^2&\text{y}^2&\text{z}^2\\\text{x}^4&\text{y}^4&\text{z}^4 \end{vmatrix}=\begin{vmatrix}\text{x}^2&\text{y}^2&\text{z}^2\\\text{x}^4&\text{y}^4&\text{z}^2\\\text{x}&\text{y}&\text{z}\end{vmatrix}$
$=\text{xyz}(\text{x}-\text{y})(\text{y}-\text{z})(\text{z}-\text{x})(\text{x}+\text{y}+\text{z}).$

Prove that the curves $\text{y}^{2} = 4x \text{ and } x^{2} = 4y $ divided the area of square bounded by $x = 0, x = 4,y=4 \text{ and } y = 0$ into three equal parts.

Find the equation of the plane passing through (a, b, c) and parallel to the plane $\vec{\text{r}}\cdot(\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}})=2.$

Evaluate the following integrals:
$\int\limits^{\pi}_0\text{x}\sin\text{x}\cos^2\text{x dx}$

Solve the following systems of linear equations by cramer's rule:

Evalute the following integrals:
$\int\frac{\text{cosec x}}{\log\tan\frac{\text{x}}{2}}\text{dx}$