‹ Back to the paper
Solve the following.
Evaluate: ∫ sin³ x cos⁴ x dx.
Evaluate: $\int \sin^3 x \cos^4 x \, dx$.
Answer
Answer
Integral: $\int \left(sin(x)^3 cos(x)^4\right) dx$
$\int\sin^3 x\cos^4 x\,dx = \int\sin^2 x\cos^4 x\,\sin x\,dx = \int(1 - \cos^2 x)\cos^4 x\,\sin x\,dx$
Put $t = \cos x$, $dt = -\sin x\,dx$:
$= -\int(t^4 - t^6)\,dt = -\frac{t^5}{5} + \frac{t^7}{7} + C$
Hence $\int\sin^3 x\cos^4 x\,dx = \frac{\cos^7 x}{7} - \frac{\cos^5 x}{5} + C$.
Final answer: $\frac{\cos^7 x}{7} - \frac{\cos^5 x}{5} + C$