Questions: ∫(cos x-3 x²) dx

Transcript text: $\int\left(\cos x-3 x^{2}\right) d x$

Solution

Solution Steps

To solve the integral of the function $\cos x - 3x^2$, we can split the integral into two separate integrals: one for $\cos x$ and another for $-3x^2$. We then integrate each part separately using standard integration techniques.

Step 1: Set Up the Integral

We start with the integral $\int\left(\cos x - 3x^2\right) dx$. This can be separated into two integrals: \[ \int \cos x \, dx - \int 3x^2 \, dx \]

Step 2: Integrate Each Term

We compute the integrals separately:

The integral of $\cos x$ is $\sin x$.
The integral of $-3x^2$ is $-x^3$.

Thus, we have: \[ \int \cos x \, dx = \sin x \] \[ \int -3x^2 \, dx = -x^3 \]

Step 3: Combine the Results

Combining the results from the two integrals, we get: \[ \int\left(\cos x - 3x^2\right) dx = \sin x - x^3 + C \] where $C$ is the constant of integration.

Final Answer

The final result of the integral is: \[ \boxed{\sin x - x^3 + C} \]

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