Evaluate the following integrals: (3 -2) 13- ^2x-7 dx - Vidyadip

Question

Evaluate the following integrals:$\int\frac{(3\sin\text{x}-2)\cos\text{x}}{13-\cos^2\text{x}-7\sin\text{x}}\text{ dx}$

✓

Answer

$\text{I}=\int\frac{(3\sin\text{x}-2)\cos\text{x}}{13-\cos^2\text{x}-7\sin\text{x}}\text{ dx}$ $=\int\frac{(3\sin\text{x}-2)\cos\text{x}}{13-(1-\sin^2\text{x})-7\sin\text{x}}\text{ dx}$ $\big(\because\ \cos^2\text{x}=1-\sin^2\text{x}\big)$ $=\int\frac{(3\sin\text{x}-2)\cos\text{x}}{\sin^2\text{x}-7\sin\text{x}+12}\text{ dx}$ $=\int\frac{(3\sin\text{x}-2)\cos\text{x}}{\sin^2\text{x}-4\sin\text{x}-3\sin\text{x}+12}\text{ dx}$ $=\int\frac{(3\sin\text{x}-2)\cos\text{x}}{\sin\text{x}(\sin\text{x}-4)-3(\sin\text{x}-4)}\text{dx}$ $=\int\frac{(3\sin\text{x}-2)\cos\text{x}}{(\sin\text{x}-3)(\sin\text{x}-4)}\text{ dx}$Let $\sin\text{x}=\text{t}$
$\Rightarrow\cos\text{x}\text{ dx}=\text{dt}$
$\therefore\ \text{I}=\int\frac{(3\text{t}-2)}{(\text{t}-3)(\text{t}-4)}\text{ dt}$
Using partial fraction, we get $\frac{(3\text{t}-2)}{(\text{t}-3)(\text{t}-4)}=\frac{\text{A}}{(\text{t}-3)}+\frac{\text{B}}{(\text{t}-4)}$ $=\frac{\text{A}(\text{t}-4)+\text{B}(\text{t}-3)}{(\text{t}-3)(\text{t}-4)}$ $\Rightarrow3\text{t}-2=(\text{A}+\text{B})\text{t}-4\text{A}-3\text{B}$ Comparing coefficients, we get A = -7 and B = 10 So, $\text{I}=-7\int\frac{1}{(\text{t}-3)}\text{ dt}+10\int\frac{1}{(\text{t}-4)}\text{ dt}$$\Rightarrow\text{I}=-7\ln|\text{t}-3|+10\ln|\text{t}-4|+\text{C}$
$\therefore\ \text{I}=-7\ln|\sin\text{x}-3|+10\ln|\sin\text{x}-4|+\text{C}$

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 "Solve the Following Question.(5 Marks)" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Find the equation of the perpendicular drawn from the point P(2, 4, -1) to the line $\frac{\text{x}+5}{1}=\frac{\text{y}+3}{4}=\frac{\text{z}-6}{-9}.$ Also, write down the coordinates of the foot of the perpendicular from P.

Find the maximum and minimum values of the function $f(x)=\cos ^2 x+\sin x$.

Evaluate:
$\begin{vmatrix}\text{a}&\text{b}&\text{c}\\\text{c}&\text{a}&\text{b}\\\text{b}&\text{c}&\text{a}\end{vmatrix}$

Find the equation of the plane that bisects the line segment joining the points (1, 2, 3) and (3, 4, 5) and is at right angle to it.

verify that $\text{y}=\text{-x}-1$ is a solution of the differential equation $(\text{y}-\text{x})\text{dy}-(\text{y}^2-\text{x}^2)\text{dx}=0.$

Evaluate : $\int \sqrt{\frac{8-x}{x}} \cdot d x$

Evaluate the following integrals:$\int^\limits{\frac{\pi}{6}}_{0}\cos^{-3}2\theta\sin2\theta\text{ d}\theta$

The management committee of a residential colony decided to award some of its members (say x) for honesty, some (say y) for helping others and some others (say z) for supervising the workers to keep the colony neat and clean. The sum of all the awardees is 12. Three times the sum of awardees for cooperation and supervision added to two times the number of awardees for honesty is 33. If the sum of the number of awardees for honesty and supervision is twice the number of awardees for helping others, using matrix method, find the number of awardees of each category. Apart from these values, namely, honesty, cooperation and supervision, suggest one more value which the management of the colony must include for awards.

Solve the matrix equations:
$\begin{bmatrix}1&2&1\end{bmatrix}\begin{bmatrix}1&2&0\\2&0&1\\1&0&2\end{bmatrix}\begin{bmatrix}0\\2\\\text{x}\end{bmatrix}=0$

Show that the height of a closed right circular cylinder of given volume and least surface area is equal to its diameter.