Question

Integrate the following expressions.
(integral) (2x^3-3x^2+5x-2)dx
(integral) (5sin2+2cos4)dx

Answer 1

Answer on Question #52848 – Math – Integral Calculus

Integrate the following expressions:

a) $\int (2x^3 - 3x^2 + 5x - 2)dx$;

b) $\int (5\sin 2 + 2\cos 4)dx$.

Solution

a) $\int (2x^3 - 3x^2 + 5x - 2)dx = \frac{1}{2} x^4 - x^3 + \frac{5}{2} x^2 - 2x + C$, where $C$ is an arbitrary real constant.

Here the following formulas were used:

$\int (kf(x))dx = k\int f(x)dx, k$ is a fixed real constant;

$$\int (f(x) + g(x))dx = \int f(x)dx + \int g(x)dx,$$ $$\int (f(x) - g(x))dx = \int f(x)dx - \int g(x)dx,$$

$\int x^n dx = \frac{x^{n+1}}{n+1} + C$, where $C$ is an arbitrary real constant, $n$ is integer, $n \neq -1$.

b) $\int (5\sin 2 + 2\cos 4)dx = (5\sin 2 + 2\cos 4)x + C$, where $C$ is an arbitrary real constant.

Here the following formula was used: $\int kdx = kx + C$, where $C$ is an arbitrary real constant, $k$ is a fixed real constant.

$$\int (5\sin 2x + 2\cos 4x)dx = \left(-\frac{5}{2}\cos 2x + \frac{2}{4}\sin 4x\right) + C = -\frac{5}{2}\cos 2x + \frac{1}{2}\sin 4x + C,$$

where $C$ is an arbitrary real constant.

Here the following formulas were used:

$\int (kf(x))dx = k\int f(x)dx, k$ is a fixed real constant;

$$\int (f(x) + g(x))dx = \int f(x)dx + \int g(x)dx,$$ $$\int \sin(ax)dx = -\frac{\cos(ax)}{a} + C,$$

$\int \cos(bx)dx = \frac{\sin(bx)}{b} + C$, where $C$ is an arbitrary real constant, $a, b$ are fixed real constants.

Answer:

a) $\int (2x^3 - 3x^2 + 5x - 2)dx = \frac{1}{2} x^4 - x^3 + \frac{5}{2} x^2 - 2x + C$;

b) $\int (5\sin 2 + 2\cos 4)dx = (5\sin 2 + 2\cos 4)x + C$.

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