Open Math
Further calculus skills Calculus · ME-12-04
Integration by substitution Q1 Straightforward
To evaluate ∫ 2 x ( x 2 + 3 ) 4 d x \displaystyle\int 2x(x^2+3)^4\,dx ∫ 2 x ( x 2 + 3 ) 4 d x using the substitution u = x 2 + 3 u = x^2 + 3 u = x 2 + 3 , what is d u du d u in terms of d x dx d x ? A. d u = 2 x d x du = 2x\,dx d u = 2 x d x B. d u = x d x du = x\,dx d u = x d x C. d u = 2 d x du = 2\,dx d u = 2 d x D. d u = ( x 2 + 3 ) d x du = (x^2+3)\,dx d u = ( x 2 + 3 ) d x
Q2 Straightforward
Using the substitution u = 3 x + 1 u = 3x + 1 u = 3 x + 1 , evaluate ∫ ( 3 x + 1 ) 5 d x \displaystyle\int (3x+1)^5\,dx ∫ ( 3 x + 1 ) 5 d x . Which of the following is correct? A. ( 3 x + 1 ) 6 18 + C \dfrac{(3x+1)^6}{18} + C 18 ( 3 x + 1 ) 6 + C B. ( 3 x + 1 ) 6 6 + C \dfrac{(3x+1)^6}{6} + C 6 ( 3 x + 1 ) 6 + C C. ( 3 x + 1 ) 6 3 + C \dfrac{(3x+1)^6}{3} + C 3 ( 3 x + 1 ) 6 + C D. 5 ( 3 x + 1 ) 4 + C 5(3x+1)^4 + C 5 ( 3 x + 1 ) 4 + C
Q3 Straightforward
Evaluate ∫ 0 1 2 x ( x 2 + 1 ) 3 d x \displaystyle\int_0^1 2x(x^2+1)^3\,dx ∫ 0 1 2 x ( x 2 + 1 ) 3 d x . Give your answer as a fraction in the form p / q p/q p / q (enter the decimal equivalent to 2 decimal places).
Q4 Straightforward
Using the substitution u = x 3 − 2 u = x^3 - 2 u = x 3 − 2 , the integral ∫ 3 x 2 e x 3 − 2 d x \displaystyle\int 3x^2 e^{x^3 - 2}\,dx ∫ 3 x 2 e x 3 − 2 d x becomes: A. ∫ e u d u \displaystyle\int e^u\,du ∫ e u d u B. ∫ 3 x 2 e u d u \displaystyle\int 3x^2 e^u\,du ∫ 3 x 2 e u d u C. ∫ e u ⋅ 3 x 2 d u \displaystyle\int e^u \cdot 3x^2\,du ∫ e u ⋅ 3 x 2 d u D. ∫ e u 3 x 2 d u \displaystyle\int \frac{e^u}{3x^2}\,du ∫ 3 x 2 e u d u
Q5 Moderate
Using the substitution u = sin x u = \sin x u = sin x , find ∫ cos x ⋅ sin 4 x d x \displaystyle\int \cos x \cdot \sin^4 x\,dx ∫ cos x ⋅ sin 4 x d x in the form sin n x k + C \dfrac{\sin^n x}{k} + C k sin n x + C . What is the value of k k k ?
Q6 Moderate
Evaluate ∫ 1 e ln x x d x \displaystyle\int_1^e \frac{\ln x}{x}\,dx ∫ 1 e x ln x d x . Give your answer to 2 decimal places.
Q7 Moderate
Use the substitution u = 4 − x 2 u = 4 - x^2 u = 4 − x 2 to find ∫ 0 1 x 4 − x 2 d x \displaystyle\int_0^1 \frac{x}{\sqrt{4-x^2}}\,dx ∫ 0 1 4 − x 2 x d x . Give your answer to 4 decimal places.
Q8 Moderate
Find ∫ x x + 1 d x \displaystyle\int x\sqrt{x+1}\,dx ∫ x x + 1 d x using the substitution u = x + 1 u = x + 1 u = x + 1 . The answer has the form 2 ( x + 1 ) 3 / 2 ( a x + b ) 15 + C \dfrac{2(x+1)^{3/2}(ax+b)}{15} + C 15 2 ( x + 1 ) 3/2 ( a x + b ) + C . What is the value of a a a ?
Q9 Moderate
Evaluate ∫ 0 π / 4 sin ( 2 x ) d x \displaystyle\int_0^{\pi/4} \sin(2x)\,dx ∫ 0 π /4 sin ( 2 x ) d x . Give your answer to 4 decimal places.
Q10 Challenging
Use the substitution u = e x + 1 u = e^x + 1 u = e x + 1 to evaluate ∫ 0 ln 3 e x ( e x + 1 ) 2 d x \displaystyle\int_0^{\ln 3} \frac{e^x}{(e^x+1)^2}\,dx ∫ 0 l n 3 ( e x + 1 ) 2 e x d x . Give your answer as a fraction (enter the decimal to 4 decimal places).
Q11 Challenging
A student claims that ∫ 0 2 x x 2 + 1 d x = ln 5 2 \displaystyle\int_0^2 \frac{x}{x^2+1}\,dx = \dfrac{\ln 5}{2} ∫ 0 2 x 2 + 1 x d x = 2 ln 5 . Use the substitution u = x 2 + 1 u = x^2 + 1 u = x 2 + 1 to verify this. What is the value of ln 5 2 \dfrac{\ln 5}{2} 2 ln 5 to 4 decimal places?
Q12 Challenging
Use the substitution u = cos x u = \cos x u = cos x to evaluate ∫ 0 π / 2 sin x cos 3 x d x \displaystyle\int_0^{\pi/2} \sin x \cos^3 x\,dx ∫ 0 π /2 sin x cos 3 x d x . Give your answer to 4 decimal places.
Standard integral forms Q13 Straightforward
Which of the following is ∫ 1 1 − x 2 d x \displaystyle\int \frac{1}{\sqrt{1-x^2}}\,dx ∫ 1 − x 2 1 d x ? A. sin − 1 x + C \sin^{-1}x + C sin − 1 x + C B. cos − 1 x + C \cos^{-1}x + C cos − 1 x + C C. tan − 1 x + C \tan^{-1}x + C tan − 1 x + C D. − 1 − x 2 + C -\sqrt{1-x^2} + C − 1 − x 2 + C
Q14 Straightforward
Which of the following is ∫ 1 1 + x 2 d x \displaystyle\int \frac{1}{1+x^2}\,dx ∫ 1 + x 2 1 d x ? A. tan − 1 x + C \tan^{-1}x + C tan − 1 x + C B. sin − 1 x + C \sin^{-1}x + C sin − 1 x + C C. ln ( 1 + x 2 ) + C \ln(1+x^2) + C ln ( 1 + x 2 ) + C D. 1 2 ln ( 1 + x 2 ) + C \dfrac{1}{2}\ln(1+x^2) + C 2 1 ln ( 1 + x 2 ) + C
Q15 Straightforward
Evaluate ∫ 0 1 1 1 + x 2 d x \displaystyle\int_0^1 \frac{1}{1+x^2}\,dx ∫ 0 1 1 + x 2 1 d x . Give your answer to 4 decimal places.
Q16 Straightforward
Use the double-angle identity cos 2 x = 1 − 2 sin 2 x \cos 2x = 1 - 2\sin^2 x cos 2 x = 1 − 2 sin 2 x to rewrite sin 2 x \sin^2 x sin 2 x . What does sin 2 x \sin^2 x sin 2 x equal in terms of cos 2 x \cos 2x cos 2 x ?
Q17 Moderate
Evaluate ∫ 0 π / 2 cos 2 x d x \displaystyle\int_0^{\pi/2} \cos^2 x\,dx ∫ 0 π /2 cos 2 x d x . Give your answer to 4 decimal places.
Q18 Moderate
Find ∫ 1 9 − x 2 d x \displaystyle\int \frac{1}{\sqrt{9-x^2}}\,dx ∫ 9 − x 2 1 d x using the standard form ∫ 1 a 2 − x 2 d x = sin − 1 ( x a ) + C \displaystyle\int \frac{1}{\sqrt{a^2-x^2}}\,dx = \sin^{-1}\!\left(\dfrac{x}{a}\right) + C ∫ a 2 − x 2 1 d x = sin − 1 ( a x ) + C . The answer is sin − 1 ( x a ) + C \sin^{-1}\!\left(\dfrac{x}{a}\right) + C sin − 1 ( a x ) + C . What is a a a ?
Q19 Moderate
Evaluate ∫ 0 2 1 4 + x 2 d x \displaystyle\int_0^{2} \frac{1}{4+x^2}\,dx ∫ 0 2 4 + x 2 1 d x using the standard form ∫ 1 a 2 + x 2 d x = 1 a tan − 1 ( x a ) + C \displaystyle\int \frac{1}{a^2+x^2}\,dx = \dfrac{1}{a}\tan^{-1}\!\left(\dfrac{x}{a}\right) + C ∫ a 2 + x 2 1 d x = a 1 tan − 1 ( a x ) + C . Give your answer to 4 decimal places.
Q20 Moderate
Evaluate ∫ 0 π sin 2 x d x \displaystyle\int_0^{\pi} \sin^2 x\,dx ∫ 0 π sin 2 x d x . Give your answer to 4 decimal places.
Q21 Moderate
Find ∫ 3 1 − x 2 d x \displaystyle\int \frac{3}{\sqrt{1-x^2}}\,dx ∫ 1 − x 2 3 d x . The answer is k sin − 1 x + C k\sin^{-1}x + C k sin − 1 x + C . What is k k k ?
Q22 Challenging
Evaluate ∫ 0 1 / 2 1 1 − x 2 d x \displaystyle\int_0^{1/2} \frac{1}{\sqrt{1-x^2}}\,dx ∫ 0 1/2 1 − x 2 1 d x . Give your answer to 4 decimal places.
Q23 Challenging
Find ∫ 2 x + 1 1 + x 2 d x \displaystyle\int \frac{2x+1}{1+x^2}\,dx ∫ 1 + x 2 2 x + 1 d x . The answer has the form ln ( 1 + x 2 ) + k tan − 1 x + C \ln(1+x^2) + k\tan^{-1}x + C ln ( 1 + x 2 ) + k tan − 1 x + C . What is k k k ?
Q24 Challenging
Evaluate ∫ 0 π / 4 sin 2 x d x \displaystyle\int_0^{\pi/4} \sin^2 x\,dx ∫ 0 π /4 sin 2 x d x . Give your answer to 4 decimal places.
Worked solutions and answers at openmath.au/year-12/extension-1/further-calculus-skills