Question

1 Evaluate
∫e4xdx

14e4x+c

ex+c

3ex3+c

13ex+c

2 Evaluate
∫xe6xdx

x6e6x−1136e6x+c

x3e6x+116e6x+c

x6e6x+1136e6x+c

−x6e6x+1136e6x+c

3 Find
∫xcosax2dx
with respect to x

cos3x+c

sin2x+c

sec2x+1

12asinax2+c

Answer 1

Answer on Question #61960 – Math – Calculus

Question

1. Evaluate $\int e^{4x} dx$.

14e4x+c

ex+c

3ex3+c

13ex+c

Solution

Using the substitution method we obtain

$$\int e^{4x} dx = \begin{cases} 4x = y, & x = \frac{y}{4} \\ dx = \frac{1}{4} dy \end{cases} = \int \frac{e^y}{4} dy = \frac{e^y}{4} + c = \frac{e^{4x}}{4} + c,$$

where $c$ is an integration constant.

Answer: $\frac{e^{4x}}{4} + c$.

Question

2. Evaluate $\int x e^{6x} dx$.

x6e6x-1136e6x+c

x3e6x+116e6x+c

x6e6x+1136e6x+c

-x6e6x+1136e6x+c

Solution

Using integration by parts we get

$$\int x e^{6x} dx = \begin{cases} u = x, & du = dx \\ dv = e^{6x} dx, & v = \frac{e^{6x}}{6} \end{cases} = \frac{x}{6} e^{6x} - \int \frac{e^{6x}}{6} dx = \frac{x}{6} e^{6x} - \frac{e^{6x}}{36} + c,$$

where $c$ is an integration constant.

Answer: $\frac{x}{6} e^{6x} - \frac{e^{6x}}{36} + c$.

Question

3. Find $\int x \cos(ax^2) \, dx$ with respect to $x$

$\cos 3x + c$

$\sin 2x + c$

$\sec 2x + 1$

$12 \text{asin} \times 2 + c$

Solution

If $a \neq 0$, then using the substitution method we obtain

$$\begin{aligned} \int x \cos(ax^2) \, dx &= \frac{1}{2a} \int 2ax \cos(ax^2) \, dx = \frac{1}{2a} \int \cos(ax^2) \, d(ax^2) = \{a x^2 = y\} = \\ &= \frac{1}{2a} \int \cos y \, dy = \frac{\sin y}{2a} + c = \frac{\sin(ax^2)}{2a} + c, \end{aligned}$$

where $a \neq 0$; $c$ is an integration constant.

If $a = 0$, then $\cos(ax^2) = \cos(0) = 1$ and

$$\int x \cos(ax^2) \, dx = \int x \, dx = \frac{x^2}{2} + c,$$

where $a = 0$; $c$ is an integration constant.

Answer: $\frac{\sin(ax^2)}{2a} + c, a \neq 0$.

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