Evaluate $\displaystyle\int \sin^5{(x)}\cos^4{(x)} \ dx$
How will I go about evaluating the given integral? I have absolutely no idea where to start, since substitution does not seem to lead anywhere.
$\endgroup$ 35 Answers
$\begingroup$Write $u = \cos x$. Then $du = -\sin x \ dx$ and the integral
$$\int \sin^5 x \cos^4 x \ dx = -\int (1 - u^2)^2 u^4 \ du$$
Now you just have to integrate a polynomial in $u$.
$\endgroup$ $\begingroup$Hint: make the change of variables $u = \cos x$, using $$ \sin^5 x = \sin x (1-\cos^2 x)^2 $$
$\endgroup$ $\begingroup$Hint: $\sin^5x = \sin x\cdot (1-\cos^2x)^2$, and use $u = \cos x$
$\endgroup$ $\begingroup$Hint:
Let $u=\cos(x)$. Then find $u'$, and substitute in $du = u' dx$. . Also, rewrite $\sin^4(x)$ as $(\sin^2(x))^2$, then as $((1-\cos^2(x))^2$.
$\endgroup$ $\begingroup$Hint: Use $$u=\cos(x), du=-\sin(x) $$ And also use $$\sin^2(x)=1-\cos^2(x)$$
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