Answer to Question #346625 in Calculus for Main

Question

1. ∫ (𝑥²+5)³ 2𝑥𝑑𝑥

2. ∫ 12𝑥² √4𝑥³ + 7𝑑𝑥

3. ∫ √2𝑥³+7 𝑥²𝑑𝑥

4. ∫ 5𝑥√1 + 4𝑥²

5. ∫(3𝑥² − 4𝑥 + 2)² (3𝑥 − 2)𝑑𝑥

Answer 1

1.

$$\int(x^2+5)^32xdx$$

$u=x^2+5, du=2xdx$

$$\int(x^2+5)^32xdx=\int u^3du=\dfrac{u^4}{4}+C$$ $$=\dfrac{(x^2+5)^4}{4}+C$$

2.

$$\int12x^2\sqrt{4x^3+7}dx$$

$u=4x^3+7, du=12x^2dx$

$$\int12x^2\sqrt{4x^3+7}dx=\int\sqrt{u}du=\dfrac{2}{3}u^{3/2}+C$$ $$=\dfrac{2}{3}(4x^3+7)^{3/2}+C$$

3.

$$\int(\sqrt{2x^3+7})x^2dx$$

$u=2x^3+7, du=6x^2dx$

$$\int(\sqrt{2x^3+7})x^2dx=\dfrac{1}{6}\int\sqrt{u}du=\dfrac{1}{9}u^{3/2}+C$$ $$=\dfrac{1}{9}(2x^3+7)^{3/2}+C$$

4.

$$\int5x\sqrt{1+4x^2}dx$$

$u=1+4x^2, du=8xdx$

$$\int5x\sqrt{1+4x^2}dx=\dfrac{5}{8}\int\sqrt{u}du=\dfrac{5}{12}u^{3/2}+C$$ $$=\dfrac{5}{12}(1+4x^2)^{3/2}+C$$

5.

$$\int(3x^2-4x+2)^2(3x-2)xdx$$

$u=3x^2-4x+2, du=(6x-4)dx$

$$\int(3x^2-4x+2)^2(3x-2)xdx=\dfrac{1}{2}\int u^2du$$ $$=\dfrac{u^3}{6}+C=\dfrac{(3x^2-4x+2)^3}{6}+C$$