Evaluate the following integrals: ^3x dx - Vidyadip

Question

Evaluate the following integrals:$\int\text{cosec}^3\text{x dx}$

✓

Answer

Let $\text{I}=\int\text{cosec}^3\text{dx}$
$=\int\text{cosec x}-\text{cosec}^2\text{x dx}$
using integration by parts,
$=\text{cosec x}\times\int\text{cosec}^2\text{x dx}+\int(\text{cosec x}\cot\text{x}\int\text{cosec}^2\text{x dx})\text{dx}$
$=\text{cosec x}\times(-\cot\text{x})+\int\text{cosec x}\cot\text{x}(-\cot\text{x})\text{dx}$
$=-\text{cosec x}\cot\text{x}-\int\text{cosec x}\cot^2\text{x dx}$
$=-\text{cosec x}\cot \text{x}-\int\text{cosec x}(\text{cosec}^2\text{x}-1)\text{dx}$
$=-\text{cosec x}\cot\text{x}-\int\text{cosec}^3\text{x dx}+\int\text{cosec x dx}$
$\text{I}=-\text{cosec x}\cot\text{x}-\text{I}+\log\Big|\tan\frac{\text{x}}{2}\Big|+\text{C}_1$
$2\text{I}=-\text{cosec x}\cot\text{x}+\log\Big|\tan\frac{\text{x}}{2}\Big|+\text{C}_1$
$\text{I}=-\frac{1}{2}\text{cosec x}\cot\text{x}+\frac{1}{2}\log\Big|\tan\frac{\text{x}}{2}\Big|+\text{C}$

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 "5 Marks Questions" from INTEGRALS→INTEGRALS — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

A die is thrown 6 times. If ‘getting an odd number’ is a success, what is the probability of

5 successes?
at least 5 successes?
at most 5 successes?

If $y=x^x+x^a+a^x+a^a$ then find $\frac{d y}{d x}$.

Find the absolute maximum and the absolute minimum value of the following functions in the given intervals:
$f(x) = 3x^4 - 8x^3 + 12x^2 - 48x + 25$ in $[0,3]$

Evaluate the following integrals:
$\int\frac{\text{x}}{\text{x}^4+2\text{x}^2+3}\text{dx}$

If f'(x) = a sin x + b cos x and f'(0) = 4, f(0) = 3, $\text{f}\Big(\frac{\pi}{2}\Big)=5$, find f(x).

A farmer mixes two brands P and Q of cattle feed. Brand P, costing Rs. 250 per bag, contains 3 units of nutritional element A, 2.5 units of element B and 2 units of element C. Brand Q costing Rs. 200 per bag contains 1.5 units of nutritional element A, 11.25 units of element B and 3 units of element C. The minimum requirements of nutrients A, B and C are 18 units, 45 units and 24 units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag? What is the minimum cost of the mixture per bag?

Evaluate the following integrals:
$\int\sec^4\text{x}\tan\text{x}\text{ dx}$

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{\text{x}}\{\log\text{y}-\log\text{x}+1\}$

Sketch the graph y = |x + 1|. Evaluate $\int\limits_{-4}^{2}|\text{x}+1|\text{dx} $ . What does this value of the integral represent on the graph.

Experiments show that radium disintegrates at a rate proportional to the amount of radium present at the moment. Its half - life is 1590 years. What percentage will disappear in one year?