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Standard integral forms

Integrate to inverse-trigonometric and logarithmic standard forms; evaluate integrals of sin⁡2x\sin^2 x and cos⁡2x\cos^2 x using double-angle identities; recognise and apply the standard results ∫1a2−x2 dx\int \frac{1}{\sqrt{a^2-x^2}}\,dx and ∫1a2+x2 dx\int \frac{1}{a^2+x^2}\,dx.

Worked examples

Inverse-sine form

Straightforward

Problem

Find ∫525−x2 dx\displaystyle\int \frac{5}{\sqrt{25-x^2}}\,dx.

Inverse-tangent form (definite integral)

Moderate

Problem

Evaluate ∫0319+x2 dx\displaystyle\int_0^3 \frac{1}{9+x^2}\,dx.

Double-angle substitution for cos⁡2x\cos^2 x

Challenging

Problem

Evaluate ∫0π/3cos⁡2x dx\displaystyle\int_0^{\pi/3} \cos^2 x\,dx.

Practise

Q1·Straightforward
Which of the following is ∫11−x2 dx\displaystyle\int \frac{1}{\sqrt{1-x^2}}\,dx?
Q2·Straightforward
Which of the following is ∫11+x2 dx\displaystyle\int \frac{1}{1+x^2}\,dx?
Q3·Straightforward
Evaluate ∫0111+x2 dx\displaystyle\int_0^1 \frac{1}{1+x^2}\,dx. Give your answer to 4 decimal places.
Q4·Straightforward
Use the double-angle identity cos⁡2x=1−2sin⁡2x\cos 2x = 1 - 2\sin^2 x to rewrite sin⁡2x\sin^2 x. What does sin⁡2x\sin^2 x equal in terms of cos⁡2x\cos 2x?
Q5·Moderate
Evaluate ∫0π/2cos⁡2x dx\displaystyle\int_0^{\pi/2} \cos^2 x\,dx. Give your answer to 4 decimal places.
Q6·Moderate
Find ∫19−x2 dx\displaystyle\int \frac{1}{\sqrt{9-x^2}}\,dx using the standard form ∫1a2−x2 dx=sin⁡−1 ⁣(xa)+C\displaystyle\int \frac{1}{\sqrt{a^2-x^2}}\,dx = \sin^{-1}\!\left(\dfrac{x}{a}\right) + C. The answer is sin⁡−1 ⁣(xa)+C\sin^{-1}\!\left(\dfrac{x}{a}\right) + C. What is aa?
Q7·Moderate
Evaluate ∫0214+x2 dx\displaystyle\int_0^{2} \frac{1}{4+x^2}\,dx using the standard form ∫1a2+x2 dx=1atan⁡−1 ⁣(xa)+C\displaystyle\int \frac{1}{a^2+x^2}\,dx = \dfrac{1}{a}\tan^{-1}\!\left(\dfrac{x}{a}\right) + C. Give your answer to 4 decimal places.
Q8·Moderate
Evaluate ∫0πsin⁡2x dx\displaystyle\int_0^{\pi} \sin^2 x\,dx. Give your answer to 4 decimal places.
Q9·Moderate
Find ∫31−x2 dx\displaystyle\int \frac{3}{\sqrt{1-x^2}}\,dx. The answer is ksin⁡−1x+Ck\sin^{-1}x + C. What is kk?
Q10·Challenging
Evaluate ∫01/211−x2 dx\displaystyle\int_0^{1/2} \frac{1}{\sqrt{1-x^2}}\,dx. Give your answer to 4 decimal places.
Q11·Challenging
Find ∫2x+11+x2 dx\displaystyle\int \frac{2x+1}{1+x^2}\,dx. The answer has the form ln⁡(1+x2)+ktan⁡−1x+C\ln(1+x^2) + k\tan^{-1}x + C. What is kk?
Q12·Challenging
Evaluate ∫0π/4sin⁡2x dx\displaystyle\int_0^{\pi/4} \sin^2 x\,dx. Give your answer to 4 decimal places.