I. $$\int 7\sin(x) dx = -7\cos(x) + C$$
II. $$\int 7\sin(x) dx = 7\cos(x) + C$$
III. $$\int 7\sin(x) dx = -\cos(x) + C$$
Which of the statements above is true?
Integrating trigonometric and inverse-trigonometric functions.
23 questions
AP Learning Target(s): LT 8: Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation
I. $$\int 7\sin(x) dx = -7\cos(x) + C$$
II. $$\int 7\sin(x) dx = 7\cos(x) + C$$
III. $$\int 7\sin(x) dx = -\cos(x) + C$$
Which of the statements above is true?
III only
II only
I only
II and III
Which of the following is an antiderivative of $$3\sec^2 x + 2$$?
$$3\sec x + 2x$$
$$\sec^3 x + 2x$$
$$3\tan x$$
$$3\tan x + 2x$$
Which of the following is an antiderivative of the function $$f(x) = \sin(2x)\cos(3x)$$?
$$-\frac{1}{10}\cos(5x) + \frac{1}{2}\cos(x) + C$$
$$\frac{1}{2}\sin(5x) - \frac{1}{2}\sin(x) + C$$
$$-\frac{1}{5}\cos(5x) + \frac{1}{4}\cos(x) + C$$
$$\frac{1}{2}\cos(5x) - \frac{1}{2}\cos(x) + C$$
Which of the following is an example of correctly finding the antiderivative of $$\cos(x)$$?
Antiderivative = $$-\cos(x)+C$$
Antiderivative = $$\sin(x)+C$$
Antiderivative = $$-\sin(x)+C$$
Antiderivative = $$\cos(x)+C$$
$$\displaystyle\int_{0}^{\pi/2} \sin(2x) dx = $$
$$0$$
$$2$$
$$1$$
$$\displaystyle\frac{1}{2}$$
Evaluate $$\int_{0}^{\pi/2} \cos(x) dx$$.
$$\sin(\pi/2) - \cos(0)$$
$$\cos(\pi/2)$$
$$\frac{\pi}{2}$$
$$1$$
Evaluate the definite integral $$\int_{0}^{\frac{\pi}{4}} \sec^2(x) dx$$ using the Fundamental Theorem of Calculus.
$$0$$
$$\frac{\pi}{4}$$
$$1$$
$$\frac{1}{2}$$
What is the value of $$\displaystyle\int_{0}^{\pi/4} \sec^2(x) dx$$?
$$\displaystyle\frac{\pi}{4}$$
$$1$$
$$\displaystyle\frac{1}{2}$$
$$\sqrt{2}$$
What is the value of $\displaystyle\int_{0}$^π sin(x)dx?
0
2
1
π
Evaluate $$\int_{0}^{\pi/2} \cos(x)\sin(x) dx$$.
$$0$$
$$\frac{\pi}{4}$$
$$\frac{1}{2}$$
$$1$$
Evaluate $\displaystyle\int_{1}$^4 (x²+1)⁻¹ dx.
ln(17/2)
π/4
tan⁻¹(4) - tan⁻¹(1)
1/2[tan⁻¹(16) - tan⁻¹(1)]
What is the value of $$\int_{\frac{\pi}{3}}^{\pi} \cos(x) dx$$?
0
-1
$$\frac{\sqrt{3}}{2}$$
$$-\frac{\sqrt{3}}{2}$$
Find an antiderivative (indefinite integral) of $$\cos(x)$$. Make sure to include the constant of integration.
$$\sin(x)+C$$
$$-\cos(x)+C$$
$$-\sin(x)+C$$
$$\cos(x)+C$$
Find the value of x that satisfies the equation $$\int_{0}^{x} \frac{1}{\sqrt{1+t^2}} dt = \ln(2).$$
$$\sqrt{\ln(2)}$$
$$\frac{5}{4}$$
$$\ln(2)$$
$$\sinh(\ln(2))$$
Which of the following is $$\int$$arctan(x) dx?
xarctan(x) - ln(x²+1) + C
xarctan(x) - ln(1+x²)/2 + C
xarctan(x) + ln(1+x²)/2 + C
xarctan(x) - ln|1+x²|/2 + C
$$\int \arcsin(x) dx =$$
$$\displaystyle x \arcsin(x) + \sqrt{1-x^2} + C$$
$$\displaystyle\frac{1}{\sqrt{1-x^2}} + C$$
$$\displaystyle x \arcsin(x) - \sqrt{1-x^2} + C$$
$$\displaystyle\frac{x^2}{2} \arcsin(x) + \frac{x\sqrt{1-x^2}}{2} + C$$
Which integration technique is most appropriate for evaluating integrals that contain inverse trigonometric functions?
u-substitution
Trigonometric substitution
Power rule
Integration by parts
Evaluate $$\int \sin^3(x) \cos^2(x) dx$$.
$$\frac{1}{5}\sin^5(x) + C$$
$$-\frac{1}{3}\cos^3(x) + \frac{1}{5}\cos^5(x) + C$$
$$\frac{1}{5}\sin^5(x)\cos(x) + C$$
$$\frac{1}{4}\sin^4(x) - \frac{1}{6}\sin^6(x) + C$$
What is the most efficient method for evaluating the integral $$\int$$tan³(x)sec⁵(x)dx?
Partial fraction decomposition after algebraic manipulation
Integration by parts with u=tan³(x) and dv=sec⁵(x)dx
Direct application of power formulas for tangent
Rewrite using tan²(x)=sec²(x)-1 and then u-substitution
What is $$\int \sin(x)\cos(x) dx$$?
$$\sin(x)*\cos(x) + C$$
$$\frac{(\sin(x))^2}{2}$$
$$-\frac{(\cos(x))^2}{2} + C$$
$$\frac{(\sin(x))^2}{2} + C$$
What is the value of $$\int_{0}^{\pi/2} \sin(x)\cos(x) dx$$?
0
π/4
1/2
1
Which of the following describes the most efficient method for evaluating the integral $$\int$$sin³(x)cos²(x)dx?
Use sin²(x) = 1-cos²(x) and then u-substitution
Integration by parts repeatedly
Direct application of power rule
Partial fraction decomposition
Which integration technique is typically most effective for an integrand involving a trigonometric function raised to a rational power?
Trigonometric substitution
Logarithmic differentiation
Power rule
u-substitution