Download App

Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIntegrals1 Mark
Question
Find: $\int \frac{(3 \sin \phi-2) \cos \phi}{5-\cos ^{2} \phi-4 \sin \phi} d \phi$ 
✓

Answer

Let $y=\sin \phi$  $\Rightarrow d y=\cos \phi d \phi$ 
Therefore, $\int \frac{(3 \sin \phi-2) \cos \phi}{5-\cos ^{2} \phi-4 \sin \phi} d \phi$ = $\int \frac{(3 y-2) d y}{5-\left(1-y^{2}\right)-4 y}$ 
= $\int \frac{3 y-2}{y^{2}-4 y+4} d y$ 
Now, we write $\frac{3 y-2}{(y-2)^{2}}=\frac{A}{y-2}+\frac{B}{(y-2)^{2}}$ 
Therefore, 3y - 2 = A (y - 2) + B
Comparing the coefficients of y and constant term,
we get A = 3 and B = 4. for y = 2, y= 0
Therefore, the required integral is given by
$I=\int\left[\frac{3}{y-2}+\frac{4}{(y-2)^{2}}\right] d y$ = $3 \int \frac{d y}{y-2}+4 \int \frac{d y}{(y-2)^{2}}$ 
= $3 \log |y-2|+4\left(-\frac{1}{y-2}\right)+C$ 
= $3 \log |\sin \phi-2|+\frac{4}{2-\sin \phi}+C$ 
= $3 \log (2-\sin \phi)+\frac{4}{2-\sin \phi}+C$ (since, sin $\phi $ $\in$[-1,1], sin $\phi $< 2, 2 - sin $\phi $ is always positive)

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 "1 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

Evaluate $\int\frac{1-\sin\text{x}}{\cos^2\text{x}}\text{ dx}$
→
If X and Y are 2 × 2 matrices, then solve the following matrix equations for X and Y.
$2\text{X}+3\text{Y}=\begin{bmatrix}2&3\\4&0\end{bmatrix},\ 3\text{X}+2\text{Y}\begin{bmatrix}-2&2\\1&-5\end{bmatrix}$
→
Show that the function given by f(x) = 3x + 17 is increasing on R.
→
Show that the function given by f(x) = 7x – 3 is increasing on R.
→
Find the direction cosines of the following vector:

 

$6\hat{\text{i}}-2\hat{\text{j}}-3\hat{\text{k}}$

 

→
The two vectors $\hat{\text{j}} + \hat{\text{k}}$ and $3\hat{\text{i}} - \hat{\text{j}} + 4\hat{\text{k}}$ represent the two sides AB and AC, respectively of a $\triangle \text{ABC},$ Find the length of the median through A.
→
If $f(x)=x \sin x$ then find $f^{\prime}\left(\frac{\pi}{2}\right)$.
→
Show that a one – one function f : {1, 2, 3} $\to$ {1, 2, 3} must be onto.
→
Classify the following measures as scalars and vectors.
40°
→
Determine the value of ‘k’ for which the following function is continuous at x = 3:
$ \text{f(x)} = \begin{cases} \frac{(\text{x + 3)}^{2} \text{ - } 36}{\text{x - 3}} & , & \text{x}\neq 3 \\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{k}& , & \text{x = 3}\\ \end{cases}$
→