Find: (2x + 5) 10 - 4x - 3x^ 2 dx - Vidyadip

Question

Find: $\int (2x + 5) \sqrt{10 - 4x - 3x^{2}} \text{ dx}$

✓

Answer

$\text{I} = \int {(\text{2x + 5)}} \sqrt{10 - \text{4x - 3x}^{2}} \text{dx}$
$= -\frac{1}{3} \int(-4-6\text{x}) \sqrt{10 - \text{4x - 3x}^{2}} \text{dx} + \frac{11}{3} \int \sqrt{10 - \text{4x - 3x}^{2}} \text{ dx}$
$= -\frac{2}{9} (10 - \text{4x - 3x}^{2})^{3/2} + \frac{11\sqrt{3}}{3} \int \sqrt{\bigg(\frac{\sqrt{34}}{3}\bigg)^{2} - \bigg(\text{x} - \frac{2}{3}\bigg)^{2}} \text{dx}$
$= -\frac{2}{9} (10 - \text{4x - 3x}^{2})^{3/2} + \frac{11\sqrt{3}}{3}\Bigg[\frac{\big(\text{x} - \frac{2}{3}\big) - \sqrt{\big(\frac{\sqrt{34}}{3}\big)^{2} - \big(\text{x} - \frac{2}{3}\big)^{2}}}{2} + \frac{17}{9} \sin^{-1} \frac{3\text{x - 2}}{\sqrt{34}} \Bigg] + \text{C}$

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 "5 Marks Questions" 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

Evalute the following integrals:
$\int\frac{-\sin\text{x}+2\cos\text{x}}{2\sin\text{x}+\cos\text{x}}\text{dx}$

Classify the following functions as injection, surjection or bijection: $f : R \rightarrow R,$ defined by $\text{f(x)}=\frac{\text{x}}{\text{x}^2+1}$

If $\vec{\text{a}}$ and $\vec{\text{b}}$ are two non-collinear unit vectors such that $\big|\vec{\text{a}}+\vec{\text{b}}\big|=\sqrt{3},$ find $\big(2\vec{\text{a}}-5\vec{\text{b}}\big).\big(3\vec{\text{a}}+\vec{\text{b}}\big).$

A bag contains 10 balls, each marked with one of the digits from 0 to 9. If four balls are drawn successively with replacement from the bag, what is the probability that none is marked with the digit 0?

A merchant plans to sell two types of personal computers a desktop model and a portable model that will cost Rs. 25,000 and Rs. 40,000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant should stock to get maximum profit if he does not want to invest more than Rs. 70 lakhs and his profit on the desktop model is Rs. 4500 and on the portable model is Rs. 5000. Make an LPP and solve it graphically.

Solve the following differential equations:$\frac{\text{dy}}{\text{dx}}=\text{y}\tan\text{ x, y}(0)=1$

Find the maximum and minimum value of 2x + y subject to the constraints:

$\text{x}+3\text{y}\geq6,\text{x}-3\text{y}\leq3,3\text{x}+4\text{y}\leq24,$ $-3\text{x}+2\text{y}\leq6,5\text{x}+\text{y}\geq5,\text{x},\text{y}\geq0$

An urn contains 5 red and 2 black balls. Two balls are randomly drawn, without replacement. Let X represent the number of black balls drawn. What are the possible values of X? Is X a random variable? If yes, find the mean and variance of X.

If $\text{A}^\text{T}=\begin{bmatrix}3&4\\-1&2\\0&1\end{bmatrix}$ and $\text{B}=\begin{bmatrix}-1&2&1\\1&2&3\end{bmatrix},$ find $A^T - B^T$.

Evaluate the following intregals:
$\int\frac{1}{(\sin\text{x}-2\cos\text{x})(2\sin\text{x}-\cos\text{x})}\ \text{dx}$