Evaluate the following intregals: (1- )(2- ) dx - Vidyadip

Question

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

✓

Answer

We have,
$\text{I}=\int\frac{\cos\text{x}\text{ dx}}{(1-\sin\text{x})(2-\sin\text{x})}$
putting $\sin\text{x}=\text{t}$
$\Rightarrow\cos\text{x dx}=\text{dt}$
$\therefore\text{I}=\int\frac{\text{dt}}{(1-\text{t})(2-\text{t})}$
$\therefore\text{I}=\int\frac{\text{dt}}{(1-\text{t})(2-\text{t})}$
Let $\frac{1}{(\text{t}-1)(\text{t}-2)}=\frac{\text{A}}{\text{t}-1}+\frac{ \text{B}}{\text{t}-2}$
$\Rightarrow\frac{1}{(\text{t}-1)(\text{t}-2)}=\frac{\text{A}(\text{t}-2)+\text{B}(\text{t}-1)}{(\text{t}-1)(\text{t}-2)}$
$\Rightarrow1=\text{A}(\text{t}-2)+\text{B}(\text{t}-1)$
Putting t - 1 = 0
⇒ t = 1
$\therefore$ 1 = A (1 - 2) + B × 0
⇒ A = -1
Putting t - 2 = 0
⇒ t = 2
$\therefore$ 1 = A × 0 + B(2 - 1)
⇒ B = 1
$\therefore\text{I}=\int\frac{-\text{dt}}{\text{t}-1}+\int\frac{\text{dt}}{\text{t}-2}$
$=-\log|\text{t}-1|+\log|\text{t}-2|+\text{C}$
$=\log\Big|\frac{\text{t}-2}{\text{t}-1}\Big|+\text{C}$
$=\log\Big|\frac{\sin\text{x}-2}{\sin\text{x}-1}\Big|+\text{C}$
$=\log\Big|\frac{2-\sin\text{x}}{1-\sin\text{x}}\Big|+\text{C}$

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 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

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

Determine the product and use it to $\begin{bmatrix} -4 & 4 & 4 \\ -7 & 1 & 3 \\ 5 & -3 & -1 \end{bmatrix} \begin{bmatrix} 1 & -1 & 1 \\ 1 & -2 & -2 \\ 2 & 1 & 3 \end{bmatrix}$solve the system of equations x – y + z = 4, x – 2y – 2z = 9, 2x + y + 3z = 1.

Show that four points whose position vectors are
$6\hat{\text{i}}-7\hat{\text{j}},16\hat{\text{i}}-19\hat{\text{j}}-4\hat{\text{k}},3\hat{\text{i}}-6\hat{\text{k}},2\hat{\text{i}}-5\hat{\text{j}}+10\hat{\text{k}}$ are coplanar.

If AD is the median of $\triangle\text{ABC},$ using vectors, prove that $\text{AB}^2+\text{AC}^2=2\big(\text{AD}^2+\text{CD}^2\big).$

Find the equation of the curve such that the portion of the x-axis cut off between the origin and the tangent at a point is twice the abscissa and which passes through the point (1, 2).

Solve the following differential equation
$\text{C}(\text{x})=2+0.15\text{x},\text{C}(0)=100$

If $\text{A}=\begin{bmatrix}3&-2\\4&-2\end{bmatrix}$ and $\text{I}=\begin{bmatrix}1&0\\0&1\end{bmatrix},$ then prove that A² - A + 2I = O.

Form the differential equation of all the circle which pass through the origin and whose centres lies in x-axis.

Let E₁ and E₂ be two independent events such that P(E₁) = P₁and P(E₂) = P₂. Describe in words of the events whose probabilities are:

P₁P₂
(1 - P₁)P₂
1 - (1 - P₁)(1 - P₂)
P₁ + P₂- 2P₁P₂

Solve the follwing system of equations by matrix method:
5x + 2y = 3
3x + 2y = 5