Download App

Indefinite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals5 Marks
Question
Evaluvate the following intregals:
$\int\frac{2\tan\text{x}+3}{3\tan\text{x}+4}\ \text{dx}$
✓

Answer

Let $\text{I}=\int\frac{2\tan\text{x}+3}{3\tan\text{x}+4}\ \text{dx}$
$=\int\frac{2\sin\text{x}+3\cos\text{x}}{3\sin\text{x}+4\cos\text{x}}\ \text{dx}$
Let $2\sin\text{x}+3\cos\text{x}=\lambda\frac{\text{d}}{\text{dx}}(3\sin\text{x}+4\cos\text{x})+\mu(3\sin\text{x}+4\cos\text{x})+\text{v}$
$2\sin\text{x}+3\cos\text{x}=\lambda(3\cos\text{x}-4\sin\text{x})+\mu(3\sin\text{x}+4\cos\text{x})+\text{v}$
$2\sin\text{x}+3\cos\text{x}=(3\lambda+4\mu)\cos\text{x}+(-4\lambda+3\mu)\sin\text{x}+\text{v}$
Comparing the coefficient of $\sin\text{x}\ \&\cos\text{x}$ on the both the sides,
$3\lambda+4\mu=3\ \dots\dots(1)$
$-4\lambda+3\mu=2\ \dots\dots(2)$
Solving the equation (1), (2) and (3),
$\mu=\frac{18}{25}$
$\lambda=\frac{1}{25}$
$\text{V}=0$
$\text{I}=\frac{1}{25}\int\frac{(3\cos\text{x}-4\sin\text{x})}{(3\sin\text{x}+4\cos\text{x})}\ \text{dx}+\frac{18}{25}\int\text{dx}$
$\text{I}=\frac{18}{25}\text{x}+\frac{1}{25}\log|3\sin\text{x}+4\cos\text{x}|+\text{C}$

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 Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Using integration, Find the area bounded by the the triangle whose vartices are $(2, 1), (3, 4)$ and $(5, 2)$.
→
If $\text{A}=\begin{bmatrix}2&0&1\\2&1&3\\1&-1&0\end{bmatrix},$ find $A^2 - 5A + 4I$ and hence find a matrix $X$ such that $A^2 - 5A + 4I + X = 0$.
→
Evaluate :$\int\Big(\sqrt{\cot\text{x}}+\sqrt{\tan\text{x}}\Big)\text{dx}$
→
Evaluate the following:
$\int\sqrt{\frac{\text{a}+\text{x}}{\text{a}-\text{x}}}\text{dx}$
→
Evaluate: $\int$sin x sin 2x sin 3x dx.
→
Without expanding, show that the values of the following determinant are zero:
$\begin{vmatrix}\sin\alpha&\cos\alpha&\cos(\alpha+\delta)\\\sin\beta&\cos\beta&\cos(\beta+\delta)\\\sin\gamma&\cos\gamma&\cos(\gamma+\delta)\end{vmatrix}$
→
show that the differential equation of which $\text{y}=2(\text{x}^2-1)+\text{ce}^{-\text{x}^{2}}$ is a solution, is $\frac{\text{dy}}{\text{dx}}+2\text{xy}=4\text{x}^3$
→
Solve the following differential equation:
$\text{x}\frac{\text{dy}}{\text{dx}}-\text{y}=2\sqrt{\text{y}^2-\text{x}^2}$
→
Differentiate the following functions with respect to x:
$(\sin\text{x})^{\log\text{x}}$
→
To maintain his health a person must fulfil certain minimum daily requirements for several kinds of nutrients. Assuming that there are only three kinds of nutrients-calcium, protein and calories and the person's diet consists of only two food items, I and II, whose price and nutrient contents are shown in the table below:
 
Food I
(per Ib)
Food II
(per Ib)
Minimum daliy requarement
for the nutrient
Calcium
10
5
20
Protein
5
4
20
Calories
2
6
13
Price (Rs)
60
100
  
What combination of two food items will satisfy the daily requirement and entail the least cost? Formulate this as a LPP.
→