Get the answer to "Use the methods introduced evaluate the following integrals...."

High School
Calculus and Analysis
Calculus 2
Applications of integrals

hadejada7x

Answered question

2022-01-04

Use the methods introduced evaluate the following integrals.

\int_{- 2}^{1} \frac{3}{x^{2} + 4 x + 13} d x

Answer & Explanation

Jordan Mitchell

Beginner2022-01-05Added 31 answers

Step 1
The given integral is

\int_{- 2}^{1} \frac{3}{x^{2} + 4 x + 13} d x

.
Rewrite the given integral as follows:

\int_{- 2}^{1} \frac{3}{x^{2} + 4 x + 13} d x = 3 \int_{- 2}^{1} \frac{1}{x^{2} + 4 x + 13} d x

= 3 \int_{- 2}^{1} \frac{1}{{(x + 2)}^{2} + 9}

Step 2
Apply u-substitution:

u = x + 2 \Rightarrow d u = d x

when

x = 1 \Rightarrow u = 3

when

x = - 2 \Rightarrow u = 0

Now,

\int_{- 2}^{1} \frac{3}{x^{2} + 4 x + 13} d x = 3 \int_{0}^{3} \frac{1}{u^{2} + 9} d x

Apply integral substitution: u=3v

\int_{- 2}^{1} \frac{3}{x^{2} + 4 x + 13} d x = 3 \int_{0}^{1} \frac{1}{3 (v^{2} + 1)} d x

\int_{02}^{1} \frac{3}{x^{2} + 4 x + 13} d x = \int_{0}^{1} \frac{1}{(v^{2} + 1)} d x

= {[\arctan (v)]}_{0}^{1}

= \frac{π}{4}

Terry Ray

Beginner2022-01-06Added 50 answers

\int_{- 2}^{1} \frac{3}{x^{2} + 4 x + 13} d x

We calculate the integral:

3 \int \frac{1}{{(x + 2)}^{2} + 9} d x = \arctan (\frac{1}{3} \cdot (x + 2))

Answer:

\arctan (\frac{1}{3} (x + 2)) + C

We calculate the definite integral:

\int_{- 2}^{1} \frac{3}{x^{2} + 4 x + 13} d x = (\arctan (\frac{1}{3} \cdot (x + 2))) ∣_{- 2}^{1}

F (1) = \frac{π}{4}

F(-2)=0

I = \frac{π}{4} - (0) = \frac{π}{4}

karton

Expert2022-01-11Added 613 answers

$\int_{- 2}^{1} \frac{3}{x^{2} + 4 x + 13} d x \int \frac{3}{x^{2} + 4 x + 13} d x \int \frac{3}{x^{2} + 4 x + 4 + 9} d x \int \frac{3}{(x + 2)^{2} + 9} d x 3 \times \int \frac{1}{(x + 2)^{2} + 9} d x 3 \times \int \frac{1}{t^{2} + 9} d t 3 \times \frac{1}{3} \times \arctan (\frac{t}{3}) 3 \times \frac{1}{3} \times \arctan (\frac{x + 2}{3}) \arctan (\frac{x + 2}{3}) \arctan (\frac{x + 2}{3}) |_{- 2}^{1} \arctan (\frac{1 + 2}{3}) - \arctan (\frac{- 2 + 2}{3}) A n s w e r : \frac{π}{4}$

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