Question

∫2x³ sin(5x) dx

Explanation:

Step1: Use integration by parts formula

$$\int u dv=uv-\int v du$$
Let $u = 2x^{3}$, $dv=\sin(5x)dx$. Then $du = 6x^{2}dx$, $v=-\frac{1}{5}\cos(5x)$.
$$\int 2x^{3}\sin(5x)dx=-\frac{2}{5}x^{3}\cos(5x)+\frac{6}{5}\int x^{2}\cos(5x)dx$$

Step2: Apply integration by parts again

For $\int x^{2}\cos(5x)dx$, let $u = x^{2}$, $dv=\cos(5x)dx$. Then $du = 2xdx$, $v=\frac{1}{5}\sin(5x)$.
$$\int x^{2}\cos(5x)dx=\frac{1}{5}x^{2}\sin(5x)-\frac{2}{5}\int x\sin(5x)dx$$

Step3: Apply integration by parts once more

For $\int x\sin(5x)dx$, let $u = x$, $dv=\sin(5x)dx$. Then $du = dx$, $v=-\frac{1}{5}\cos(5x)$.
$$\int x\sin(5x)dx=-\frac{1}{5}x\cos(5x)+\frac{1}{5}\int\cos(5x)dx$$

Step4: Integrate $\int\cos(5x)dx$

$$\int\cos(5x)dx=\frac{1}{5}\sin(5x)+C$$

Step5: Substitute back

Substitute the results of Step4 into Step3, then Step3 into Step2, and Step2 into Step1.
$$\int 2x^{3}\sin(5x)dx=-\frac{2}{5}x^{3}\cos(5x)+\frac{6}{25}x^{2}\sin(5x)+\frac{12}{125}x\cos(5x)-\frac{12}{625}\sin(5x)+C$$

Answer:

$$-\frac{2}{5}x^{3}\cos(5x)+\frac{6}{25}x^{2}\sin(5x)+\frac{12}{125}x\cos(5x)-\frac{12}{625}\sin(5x)+C$$