Evaluate the integrals ∫cos⁡ec40d0

Question

Talisha · Accepted Answer

Step 1
We have to evaluate the integrals:

\int \cos e c^{4} 0 d 0

Rewriting the given integral,

\int \cos e c^{4} 0 d 0 = \int \cos e c^{2} 0 \cos e c^{2} 0 d 0

\int (1 + \cot^{2} 0) \cos e c^{2} 0 d 0

Solving by substitution method,
Assuming

t = \cot 0

Differentiating,

\frac{d t}{d 0} = \frac{d (\cot 0)}{d 0}

= - \cos e c^{2} 0

- d t = \cos e c^{2} 0 d 0

Step 2
Substituting above values in the rewritten integral, we get

\int (1 + \cot^{2} 0) \cos e c^{2} 0 d 0 = \int (1 + t^{2}) (- d t)

= - \int (1 + t^{2}) d t

= - (\int d t + \int t^{2} d t)

= - t - \frac{t^{2 + 1}}{2 + 1} + C (s i n c e \int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C)

= - t - \frac{t^{3}}{3} + C

Where,C is an arbitrary constant.
Substituting back the value of t

(t = \cot 0)

, we get

\int \cos e c^{4} 0 d 0 = - t - \frac{t^{3}}{3} + C

= - \cot 0 - \frac{c o t^{3} 0}{3} + C

Hence, value of integral is -

\cot 0 - \frac{\cot^{3} 0}{3} + C

Evaluate the integrals \int \cos ec^{4}0d0