What is the integral from-a to a of |a+3t| ?

The integral from-a to a of |a+3t| is (10a|a|)/3

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integral from-a to a of |a+3t|

\int_{- a}^{a} ∣ a + 3 t ∣ d t

\frac{10 a ∣ a ∣}{3}

Solution steps

\int_{- a}^{a} ∣ a + 3 t ∣ d t

Apply u-substitution

: \int_{- 2 a}^{4 a} ∣ u ∣ \frac{1}{3} d u

= \int_{- 2 a}^{4 a} ∣ u ∣ \frac{1}{3} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{3} \cdot \int_{- 2 a}^{4 a} ∣ u ∣ d u

Use the common integral:

\int ∣ u ∣ d u = \frac{u ∣ u ∣}{2}

= \frac{1}{3} [\frac{u ∣ u ∣}{2}]_{- 2 a}^{4 a}

Compute the boundaries:

10 a ∣ a ∣

= \frac{1}{3} \cdot 10 a ∣ a ∣

Simplify

= \frac{10 a ∣ a ∣}{3}

\int_{0}^{2} r d r

\int_{0}^{3} π (x^{2})^{2} d x

\int_{3}^{7} (1 + 2 x) d x

\int_{0}^{2} (x + 3) d x

\int_{- 2}^{3} x + 3 d x

What is the integral from-a to a of |a+3t| ?
The integral from-a to a of |a+3t| is (10a|a|)/3