Evaluate the following integrals: ^ x ( -cosec^2x )dx - Vidyadip

Question

Evaluate the following integrals:$\int\text{e}^{\text{x}}\big(\cot\text{x}-\text{cosec}^2\text{x}\big)\text{dx}$

✓

Answer

Let $\text{I}=\int\text{e}^{\text{x}}\big(\cot\text{x}-\text{cosec}^2\text{x}\big)\text{dx}$
$=\int\text{e}^{\text{x}}\cot\text{x dx}-\int\text{e}^{\text{x}}\text{cosec}^2\text{x dx}$
Integration by parts
$=\text{e}^{\text{x}}\cot\text{x}-\int\text{e}^{\text{x}}\Big(\frac{\text{d}}{\text{dx}}\cot\text{x}\Big)\text{dx}-\int\text{e}^{\text{x}}\text{cosec}^2\text{x dx}$
$=\text{e}^{\text{x}}\cot\text{x}+\int\text{e}^{\text{x}}\text{cosec}^2\text{x dx}-\int\text{e}^{\text{x}}\text{cosec}^{2}\text{x dx}$
$=\text{e}^{\text{x}}\cot\text{x+C}$
$\int\text{e}^{\text{x}}\big(\cot\text{x}-\text{cosec}^2\text{x}\big)\text{dx}=\text{e}^\text{x}\cot\text{x}+\text{C}$

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 "3 Marks Question" 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

Dot product of a vector with vectore $\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}},2\hat{\text{i}}+\hat{\text{j}}-3\hat{\text{k}}$ and $\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$ are respectively 4, 0 and 2. Find the vector.

Evaluate the following integrals:$\int\frac{\log(\log\text{x})}{\text{x}}\text{dx}$

Give a condition that three vectors $\vec{\text{a}},\ \vec{\text{b}}\text{ and }\vec{\text{c}}$ from the three sides of a triangle. what are the other possibilities?

Prove that the function $f$ given by $f(x) = x - [x]$ is increasing in $(0, 1).$

Find the value of k for which the function $\text{f(x)}=\begin{cases}\frac{\text{x}^{2} + 3\text{x} - 10}{\text{x} - 2},&\text{x}\neq2\\\text{k},&\text{x} = {2}\end{cases}$ is continues at x = 2.

Find the equation of the normal to $y = 2x^3 - x^2 + 3$ at $(1, 4)$.

A pair of dice is thrown. Find the probability of getting 7 as the sum if it is known that the second die always exhibits a prime number.

The probability that at least one of the two events A and B occurs is 0.6. If A and B occur simultaneously with probability 0.3, evaluate $\text{P}\bar{(\text{A})}+\text{P}\bar{(\text{B})}.$

Let $\vec{\text{a}}=5\hat{\text{i}}-\hat{\text{j}}+7\hat{\text{k}}$ and $\vec{\text{b}}=\hat{\text{i}}-\hat{\text{j}}+\lambda\hat{\text{k}}.$ Find $\lambda$ such that $\vec{\text{a}}+\vec{\text{b}}$ is orthonal to $\vec{\text{a}}-\vec{\text{b}}.$

Verify that y = A cos x - b sin x is a solution of the differential equation.
$\frac{\text{d}^{2} \text{y}}{\text{dx}^{2}}+\text{y}=0.$