Question

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (2^(√y) dy) / 2√y

Accepted Answer

Start by examining the integral: $$\int \frac{2$$^{\sqrt{y}$$}}{2\sqrt{y}} \, dy. $$Notice that the expression involves $$\sqrt{y}$$ both in the exponent and the denominator, suggesting a substitution involving $$\sqrt{y}$$ might simplify the integral. Let $$u = \sqrt{y} = y$$^{1/2}$$. Then, differentiate both sides with respect to y$ to find du$ in terms of dy$: u$$ = $$y^{1/2}$$ \implies du = \frac{1}{2\sqrt{y}} dy$$. Rearrange the differential to express dy$ in terms of du$: from du$$ = \frac{1}{2\sqrt{y}} dy$$, multiply both sides by 2$$\sqrt{y}$$ to get dy = 2$$\sqrt{y} \, du$$. Substitute u$ and dy$ back into the integral: replace 2$$^{\sqrt{y}$$}$$ with $$2^u$$ and $$dy$$ with $$2\sqrt{y} \, du. $$Notice that the $$2\sqrt{y} in $$the denominator and numerator will cancel out, simplifying the integral to $$\int 2^u \, du. $$Now, integrate $$\int 2^u \, du$$ using the formula for integrating exponential functions with base $$a$$: $$\int a^u \, du = \frac{a^u}{\ln(a)} + C. $$Apply this formula with $$a=2 to $$find the antiderivative in terms of $$u$$, then substitute back $$u = \sqrt{y} to $$express the answer in terms of $$y.$$

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (2^(√y) dy) / 2√y

Risposta video verificata per un problema simile:

Concetti chiave

Integration by Substitution

Exponential Functions with Variable Exponents

Algebraic Manipulation of Integrands

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.∫ (2^(√y) dy) / 2√y

Risposta video verificata per un problema simile:

Concetti chiave

Integration by Substitution

Exponential Functions with Variable Exponents

Algebraic Manipulation of Integrands

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (2^(√y) dy) / 2√y