Mastering Integration Techniques

Evaluate the integral ∫(2x)sin(x^2)dx using substitution.

• 2x^2 + C

• -cos(x^2) + C

• sin(x^2) + C

• -sin(x^2) + C

Use integration by parts to solve ∫x e^x dx.

• e^x (x^2) + C

• e^x (x + 1) + C

• e^x (x - 2) + C

• e^x (x - 1) + C

Find the integral ∫(3x + 2)/(x^2 + 2x + 1)dx using partial fraction decomposition.

• 3ln|x + 1| + 1/(x + 1) + C

• ln|x^2 + 2x + 1| + C

• 2ln|x + 1| + 1/(x + 1) + C

• 3x/(x + 1) + C

Calculate ∫sin^2(x)dx using trigonometric identities.

• (1/3)x - (1/6)sin(3x) + C

• (1/4)x - (1/2)sin(2x) + C

• (1/2)x + (1/4)sin(2x) + C

• (1/2)x - (1/4)sin(2x) + C

Evaluate the integral ∫(1/(x^2 + 4))dx using trigonometric substitution.

• (1/4)arcsin(x/2) + C

• (1/4)arctan(x/2) + C

• (1/8)arctan(2x) + C

• (1/2)ln(x^2 + 4) + C

Use substitution to evaluate ∫(x^2 + 1)^(1/2)dx.

• (1/4)((x^2 + 1)^(1/2)x^2 + ln|x|) + C

• (1/2)((x^2 + 1)^(1/2)x + ln|x + (x^2 + 1)^(1/2)|) + C

• (1/2)(x^2 + 1)^(1/2) + C

• (1/3)((x^2 + 1)^(3/2) + C)

Solve ∫x^2 ln(x)dx using integration by parts.

• (1/3)x^3 ln(x) - (1/9)x^3 + C

• (1/2)x^3 ln(x) - (1/6)x^3 + C

• (1/4)x^4 ln(x) - (1/8)x^4 + C

• (1/5)x^5 ln(x) - (1/10)x^5 + C

Find the integral ∫(1/(x^2 - 1))dx using partial fraction decomposition.

• (1/2)ln|x^2 - 1| + C

• (1/2)ln|(x+1)/(x-1)| + C

• -ln|x^2 - 1| + C

• (1/2)ln|(x-1)/(x+1)| + C

Evaluate ∫cos^3(x)sin(x)dx using substitution.

• rac{sin^4(x)}{4} + C

• rac{cos^2(x)}{2} + C

• -rac{sin^3(x)}{3} + C

• -\frac{cos^4(x)}{4} + C

Use trigonometric substitution to evaluate ∫(1/(9 - x^2)^(1/2))dx.

• ln(3 - x^2) + C

• tan(x/3) + C

• cos(x/3) + C

• arcsin(x/3) + C