$$ \int _{}^{}\frac{1}{2t^{2}+5} d t $$

Question

UpStudy AI · Accepted Answer

We wish to evaluate

∫ dt/(2t² + 5).

Step 1. Write the denominator in a form that suggests a standard arctan integration formula. Notice that

2t² + 5 = 2(t²) + 5.

Step 2. Introduce a substitution to simplify the quadratic expression in t. Let

u = √2 · t  ⟹  dt = du/√2.

Then, the expression t² becomes

t² = u²/2  ⟹  2t² = u².

Thus, the denominator becomes

2t² + 5 = u² + 5.

Step 3. Substitute into the integral:

∫ dt/(2t² + 5) = ∫ (du/√2)/(u² + 5)
             = (1/√2) ∫ du/(u² + 5).

Step 4. Recognize that the standard integral

∫ du/(u² + a²) = (1/a) arctan(u/a) + C

applies here with a² = 5, which means a = √5.

Thus,

∫ du/(u² + 5) = (1/√5) arctan(u/√5) + C.

Step 5. Substitute back u = √2 · t to obtain the answer in terms of t:

(1/√2) ∙ (1/√5) arctan((√2 · t)/√5) + C = (1/√10) arctan((√2 · t)/√5) + C.

Therefore, the evaluated integral is

(1/√10) arctan((√2 · t)/√5) + C.

This is the final answer.
The integral evaluates to $$ \frac{1}{\sqrt{10}} \arctan\left(\frac{\sqrt{2} \cdot t}{\sqrt{5}}\right) + C $$.

\( \int _{}^{}\frac{1}{2t^{2}+5} d t \)