Question

Find the indefinite integral and check the result by differentiation. $$ \int \frac{t+2 t^{2}}{\sqrt{t}} d t $$

Short answer

The indefinite integral of the function \( \frac{t+2 t^{2}}{\sqrt{t}} \) is \( \frac{2}{3}t^{\frac{3}{2}} + \frac{4}{5}t^{\frac{5}{2}} + C \). The verification by differentiation proves the correctness of this result.

Step-by-step solution

Rewrite the Function

First, rewrite the given function to make it ready for integration. We can rewrite \( \frac{t+2 t^{2}}{\sqrt{t}} \) as \( t^{\frac{1}{2}} + 2t^{\frac{3}{2}} \).

Integrate

Apply the rule of integration \(\int x^n dx = \frac{1}{n+1}x^{n+1}\). So, the integral is \( \frac{2}{3}t^{\frac{3}{2}} + \frac{4}{5}t^{\frac{5}{2}} + C \), where C is the constant of integration.

Differentiate the Result

To verify, differentiate the result. The rule of differentiation \(\frac{d}{dx} x^n = nx^{n-1}\) is applied. On differentiating \( \frac{2}{3}t^{\frac{3}{2}} + \frac{4}{5}t^{\frac{5}{2}} \), we get \( \frac{1}{2}t^{-\frac{1}{2}} + 2t^{\frac{3}{2}} \), which simplifies back to the original function, thereby verifying the correctness of the integral.