∫ x² cos x dx

Q: ∫ x² cos x dx

To solve the integral $\int x^2 \cos x \, dx$, we can use integration by parts. The formula for integration by parts is: \[ \int u \, dv = uv - \int v \, du \] Let's choose $u = x^2$ and $dv = \cos x \, dx$. Then, differentiate $u$ and integrate $dv$: - $du = 2x \, dx$ - $v = \int \cos x \, dx = \sin x$ Now apply the integration by parts formula: \[ \int x^2 \cos x \, dx = x^2 \sin x - \int \sin x \cdot 2x \, dx \] The remaining integral $\int 2x \sin x \, dx$ requires integration by parts again. Choose: - $u = 2x$, $dv = \sin x \, dx$ - $du = 2 \, dx$, $v = -\cos x$ Apply the formula again: \[ \int 2x \sin x \, dx = -2x \cos x + \int 2 \cos x \, dx \] The integral $\int 2 \cos x \, dx$ is straightforward: \[ \int 2 \cos x \, dx = 2 \sin x \] Substitute back: \[ \int 2x \sin x \, dx = -2x \cos x + 2 \sin x \] Substitute this result back into the first integration by parts: \[ \int x^2 \cos x \, dx = x^2 \sin x - (-2x \cos x + 2 \sin x) \] Simplify: \[ = x^2 \sin x + 2x \cos x - 2 \sin x + C \] where $C$ is the constant of integration.

Question

∫ x² cos x dx

Accepted Answer

To solve the integral $\int x^2 \cos x \, dx$, we can use integration by parts. The formula for integration by parts is:

$$
\int u \, dv = uv - \int v \, du
$$

Let's choose $u = x^2$ and $dv = \cos x \, dx$.

Then, differentiate $u$ and integrate $dv$:

- $du = 2x \, dx$
- $v = \int \cos x \, dx = \sin x$

Now apply the integration by parts formula:

$$
\int x^2 \cos x \, dx = x^2 \sin x - \int \sin x \cdot 2x \, dx
$$

The remaining integral $\int 2x \sin x \, dx$ requires integration by parts again. Choose:

- $u = 2x$, $dv = \sin x \, dx$
- $du = 2 \, dx$, $v = -\cos x$

Apply the formula again:

$$
\int 2x \sin x \, dx = -2x \cos x + \int 2 \cos x \, dx
$$

The integral $\int 2 \cos x \, dx$ is straightforward:

$$
\int 2 \cos x \, dx = 2 \sin x
$$

Substitute back:

$$
\int 2x \sin x \, dx = -2x \cos x + 2 \sin x
$$

Substitute this result back into the first integration by parts:

$$
\int x^2 \cos x \, dx = x^2 \sin x - (-2x \cos x + 2 \sin x)
$$

Simplify:

$$
= x^2 \sin x + 2x \cos x - 2 \sin x + C
$$

where $C$ is the constant of integration.

∫ x² cos x dx

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