2- Integration by Substitution

Integration by Substitution

Multiple Choice

Use the indicated substitution to evaluate the integral.

$$\int\sin\left(2x\right)\ dx,\ \ \ \ \$$  

• $$\cos\left(2x\right)+C$$  

•   $$-\frac{1}{2}\cos\left(2x\right)+C$$  

• $$-\cos\left(2x\right)+C$$  

• $$\frac{1}{2}\cos\left(2x\right)+C$$  

Multiple Choice

Use the indicated substitution to evaluate the integral.

$$\int2x\cos(x^2)\ dx\ ,\ \$$  

• $$\sin\left(x^2\right)+C$$  

• $$2\sin\left(x^2\right)+C$$  

• $$\frac{1}{2}\sin\left(x^2\right)+C$$  

• $$4\cos\left(x^2\right)+C$$  

Multiple Choice

Use the indicated substitution to evaluate the integral.

$$\int7x^6\left(x^7+1\right)^8dx\ ,\ \$$  

• $$\frac{\left(x^7-1\right)^9}{9}+c$$  

• $$\frac{\left(x^7+1\right)^9}{9}+c$$  

• $$\frac{\left(x^7+1\right)^8}{8}+c$$  

• $$\frac{\left(7x^6+1\right)^9}{9}+c$$  

Multiple Choice

Use the indicated substitution to evaluate the integral.

$$\int\frac{2x}{\sqrt{3+x^2}}dx\ ,\ \ \ u=3+x^2$$  

• $$\frac{1}{2\sqrt{3+x^2}}+C$$  

• $$\sqrt{3+x^2}+C$$  

• $$2\sqrt{3+x^2}+C$$  

• $$\frac{2}{\sqrt{3+x^3}}+C$$  

Multiple Choice

Use substitution to evaluate the integral $$\int\left(3x^3+5\right)^5\cdot27x^2dx$$

• $$\frac{1}{2}\left(3x^3+5\right)^6+C$$

• $$\frac{2}{5}\left(3x^3+5\right)^5+C$$

• $$\frac{5}{6}\left(3x^3+5\right)^6+C$$

• $$\frac{3}{5}\left(3x^3+5\right)^5+C$$

Multiple Choice

Use substitution to evaluate the integral

$$\int\ x^2\sqrt{5+2x^3}\ dx$$  

• $$\frac{1}{9}\left(5+2x^3\right)^{\frac{3}{2}}+C$$  

• $$\frac{2}{3}\left(5+2x^3\right)^{\frac{3}{2}}+C$$  

• $$6x\left(5+2x^3\right)^{\frac{3}{2}}+C$$  

• $$\frac{1}{9}x^3\left(5+2x^3\right)^{\frac{3}{2}}+C$$  

Multiple Choice

Evaluate the integral: ∫ 2x(x² + 5)⁻⁴ dx

• (1/3)(x² + 5)⁻³ + C

• -(1/3)(x² + 5)⁻³ + C

• -4(x² + 5)⁻⁵ + C

• (x² + 5)⁻⁴ + C

Multiple Choice

Evaluate the integral: ∫ (4x³/(x⁴ + 1)²) dx

• (1/(x⁴ + 1)) + C

• -(1/(x⁴ + 1)) + C

• -2(x⁴ + 1)⁻³ + C

• ln(x⁴ + 1) + C

Multiple Choice

Evaluate the integral: ∫ (3x + 2)(3x² + 4x)⁴ dx

• (1/5)(3x² + 4x)⁵ + C

• (1/10)(3x² + 4x)⁵ + C

• (3x² + 4x)⁴ + C

• (1/4)(3x² + 4x)⁴ + C

Multiple Choice

Evaluate the integral: $$\int \sqrt{x} \sin^{2}(x^{3/2} - 1) \, dx$$

• $$(x^{3/2} - 1)/3 + C$$

• $$(x^{3/2} - 1)/3 - sin(2(x^{3/2} - 1))/6 + C$$

• - $$-cos(x^{3/2} - 1)$$ + C

• $$(2/3) sin²(x^{3/2} - 1) + C$$