Evaluate the indefinite integral. ∫ e 2 x x d x \int_{}^{}\frac{e^{2\sqrt{x}}}{\sqrt{x}}dx

we're asked to evaluate the indefinite integral of e to the two root x over root x d x. Now looking at this integral, since I have e raised to a function of x, that means that I probably need to make a substitution where u is equal to that power. So if u is equal to two root x, that then means du is equal to the derivative the derivative of that. So that's gonna be two times and then applying the power rule for derivatives here, that is one half times x to the power of negative one half, then multiplying that by d x. Now looking at this, this two and one half are going to cancel out here, and x to the power of negative one half is the same thing as one over root x. Then with that d x there, looking at what I have in my original integral, that's exactly what is there in that denominator and then, of course, with my d x. So this allows me to rewrite my integral as simply e to the u d u. And we know that when doing a substitution that results in this integral, this is just going to give us e to the u plus c. Now remember that we know what u is. It's that two root x. So we wanna go ahead and replace that. So this is e to the two root x plus c, and that's our final answer. I'll see you in the next one.

Evaluate the indefinite integral. ∫ e 2 x x d x \int_{}^{}\frac{e^{2\sqrt{x}}}{\sqrt{x}}dx

5 e 2 x 3 x 3 2 + C \frac{5e^{2\sqrt{x}}}{3x^{\frac32}}+C

3 e x 7 2 7 x 3 2 + C \frac{3e^{x^{\frac72}}}{7x^{\frac32}}+C

7 e x 7 2 3 x 3 2 + C \frac{7e^{x^{\frac72}}}{3x^{\frac32}}+C

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0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
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- Area Between Curves
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10. Physics Applications of Integrals 3h 16m
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11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
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16. Parametric Equations & Polar Coordinates7h 58m

11. Integrals of Inverse, Exponential, & Logarithmic Functions

Integrals of Exponential Functions

Calculus

11. Integrals of Inverse, Exponential, & Logarithmic Functions

Integrals of Exponential Functions

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Multiple Choice

Evaluate the indefinite integral.
$\int_{}^{} \frac{e^{2 \sqrt{x}}}{\sqrt{x}} d x$

$\frac{5 e^{2 \sqrt{x}}}{3 x^{\frac{3}{2}}} + C$

$\frac{3 e^{x^{\frac{7}{2}}}}{7 x^{\frac{3}{2}}} + C$

$e^{2 \sqrt{x}} + C$

$\frac{7 e^{x^{\frac{7}{2}}}}{3 x^{\frac{3}{2}}} + C$

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Verified step by step guidance

Step 1: Recognize that the integral involves an exponential function with a square root in the exponent. The integral is ∫(e^(2√x) / √x) dx. This suggests a substitution method is appropriate.

Step 2: Let u = 2√x. Then, differentiate u with respect to x: du/dx = (1/√x). This implies du = (1/√x) dx. Substitute this into the integral.

Step 3: After substitution, the integral becomes ∫e^u du. This is a standard integral of the exponential function.

Step 4: The integral of e^u with respect to u is simply e^u + C, where C is the constant of integration.

Step 5: Substitute back u = 2√x to return to the original variable. The final result is e^(2√x) + C.

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Struggling with Calculus?

Evaluate the indefinite integral.∫e2xxdx\int_{}^{}\frac{e^{2\sqrt{x}}}{\sqrt{x}}dx

Watch next

Integrals of Exponential Functions practice set

Evaluate the indefinite integral.
$\int_{}^{} \frac{e^{2 \sqrt{x}}}{\sqrt{x}} d x$