Evaluate the following integrals: integral 2x dx, integral e2x−4 dx, integral e3x dx, integral (2x+3)1/2 dx

Question

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Here are the solutions to the exercises:

1) Evaluate $\int \cos 2x \, dx$ .

Let $u = 2x$ .
Differentiate $u$ with respect to $x$ to find $du$ .
$\frac{du}{dx} = 2 \implies dx = \frac{1}{2} du$
Substitute $u$ and $dx$ into the integral.
$\int \cos u \cdot \frac{1}{2} du = \frac{1}{2} \int \cos u \, du$
Integrate with respect to $u$ .
$\frac{1}{2} \sin u + C$
Substitute back $u = 2x$ .
$\boxed{\frac{1}{2} \sin(2x) + C}$
2) Evaluate $\int e^{2x-4} \, dx$ .
Let $u = 2x-4$ .
Differentiate $u$ with respect to $x$ to find $du$ .
$\frac{du}{dx} = 2 \implies dx = \frac{1}{2} du$
Substitute $u$ and $dx$ into the integral.
$\int e^u \cdot \frac{1}{2} du = \frac{1}{2} \int e^u \, du$
Integrate with respect to $u$ .
$\frac{1}{2} e^u + C$
Substitute back $u = 2x-4$ .
$\boxed{\frac{1}{2} e^{2x-4} + C}$
3) Evaluate $\int e^{3x} \, dx$ .
Let $u = 3x$ .
Differentiate $u$ with respect to $x$ to find $du$ .
$\frac{du}{dx} = 3 \implies dx = \frac{1}{3} du$
Substitute $u$ and $dx$ into the integral.
$\int e^u \cdot \frac{1}{3} du = \frac{1}{3} \int e^u \, du$
Integrate with respect to $u$ .
$\frac{1}{3} e^u + C$
Substitute back $u = 3x$ .
$\boxed{\frac{1}{3} e^{3x} + C}$
4) Evaluate $\int (2x+3)^{1/2} \, dx$ .
Let $u = 2x+3$ .
Differentiate $u$ with respect to $x$ to find $du$ .
$\frac{du}{dx} = 2 \implies dx = \frac{1}{2} du$
Substitute $u$ and $dx$ into the integral.
$\int u^{1/2} \cdot \frac{1}{2} du = \frac{1}{2} \int u^{1/2} \, du$
Integrate with respect to $u$ using the power rule $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ .
$\frac{1}{2} \cdot \frac{u^{1/2+1}}{1/2+1} + C = \frac{1}{2} \cdot \frac{u^{3/2}}{3/2} + C$
Simplify the expression.
$\frac{1}{2} \cdot \frac{2}{3} u^{3/2} + C = \frac{1}{3} u^{3/2} + C$
Substitute back $u = 2x+3$ .
$\boxed{\frac{1}{3} (2x+3)^{3/2} + C}$

Evaluate the following integrals: integral 2x dx, integral e2x-4 dx, integral e3x dx, integral (2x+3)1/2 dx

Step-by-step solution

Let $u = 2x$ .

Differentiate $u$ with respect to $x$ to find $du$ .

Substitute $u$ and $dx$ into the integral.

Integrate with respect to $u$ .

Substitute back $u = 2x$ .

Let $u = 2x-4$ .

Differentiate $u$ with respect to $x$ to find $du$ .

Substitute $u$ and $dx$ into the integral.

Integrate with respect to $u$ .

Substitute back $u = 2x-4$ .

Let $u = 3x$ .

Differentiate $u$ with respect to $x$ to find $du$ .

Substitute $u$ and $dx$ into the integral.

Integrate with respect to $u$ .

Substitute back $u = 3x$ .

Let $u = 2x+3$ .

Differentiate $u$ with respect to $x$ to find $du$ .

Substitute $u$ and $dx$ into the integral.

Integrate with respect to $u$ using the power rule $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ .

Simplify the expression.

Substitute back $u = 2x+3$ .

Still stuck on this one?

More Mathematics Questions

Evaluate the following integrals: integral 2x dx, integral e2x-4 dx, integral e3x dx, integral (2x+3)1/2 dx

Step-by-step solution

Let u=2xu = 2xu=2x.

Differentiate uuu with respect to xxx to find dududu.

Substitute uuu and dxdxdx into the integral.

Integrate with respect to uuu.

Substitute back u=2xu = 2xu=2x.

Let u=2x−4u = 2x-4u=2x−4.

Differentiate uuu with respect to xxx to find dududu.

Substitute uuu and dxdxdx into the integral.

Integrate with respect to uuu.

Substitute back u=2x−4u = 2x-4u=2x−4.

Let u=3xu = 3xu=3x.

Differentiate uuu with respect to xxx to find dududu.

Substitute uuu and dxdxdx into the integral.

Integrate with respect to uuu.

Substitute back u=3xu = 3xu=3x.

Let u=2x+3u = 2x+3u=2x+3.

Differentiate uuu with respect to xxx to find dududu.

Substitute uuu and dxdxdx into the integral.

Integrate with respect to uuu using the power rule ∫xn dx=xn+1n+1+C\int x^n \, dx = \frac{x^{n+1}}{n+1} + C∫xndx=n+1xn+1​+C.

Simplify the expression.

Substitute back u=2x+3u = 2x+3u=2x+3.

Still stuck on this one?

More Mathematics Questions

Let $u = 2x$ .

Differentiate $u$ with respect to $x$ to find $du$ .

Substitute $u$ and $dx$ into the integral.

Integrate with respect to $u$ .

Substitute back $u = 2x$ .

Let $u = 2x-4$ .

Differentiate $u$ with respect to $x$ to find $du$ .

Substitute $u$ and $dx$ into the integral.

Integrate with respect to $u$ .

Substitute back $u = 2x-4$ .

Let $u = 3x$ .

Differentiate $u$ with respect to $x$ to find $du$ .

Substitute $u$ and $dx$ into the integral.

Integrate with respect to $u$ .

Substitute back $u = 3x$ .

Let $u = 2x+3$ .

Differentiate $u$ with respect to $x$ to find $du$ .

Substitute $u$ and $dx$ into the integral.

Integrate with respect to $u$ using the power rule $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ .

Substitute back $u = 2x+3$ .