Evaluate - - sin 5 x-dx

∫sin^5xdx =∫sin^4xsinxdx =∫(1 cos^2x)^2sinxdx Let u=cosx⇒ du= sinxdx = ∫(1 u^2)^2du = ∫(1+u^4 2u^2)du = [u+u^55 2u^33]+c ...

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Byju's Answer

Standard XII

Mathematics

Integration of Trigonometric Functions

Evaluate : ∫...

Question

Evaluate :
$\int {s i n}^{5} x . d x$

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Solution

$\int {sin}^{5} x d x$
$= \int {sin}^{4} x sin x d x$
$= \int {(1 - {cos}^{2} x)}^{2} sin x d x$
Let $u = cos x \Rightarrow d u = - sin x d x$
$= - \int {(1 - u^{2})}^{2} d u$
$= - \int (1 + u^{4} - 2 u^{2}) d u$
$= - [u + \frac{u^{5}}{5} - \frac{2 u^{3}}{3}] + c$
$= - cos x - \frac{{cos}^{5} x}{5} + \frac{2 {cos}^{3} x}{3} - c$
$= - \frac{{cos}^{5} x}{5} + \frac{2 {cos}^{3} x}{3} - cos x - c$

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Evaluate :∫sin5x.dx

∫sin5xdx=∫sin4xsinxdx=∫(1−cos2x)2sinxdxLet u=cosx⇒du=−sinxdx=−∫(1−u2)2du=−∫(1+u4−2u2)du=−[u+u55−2u33]+c=−cosx−cos5x5+2cos3x3−c=−cos5x5+2cos3x3−cosx−c

Evaluate :
$\int {s i n}^{5} x . d x$